The method could vary depending on your needs. Here are 3 ways to calculate Luminance:

• Luminance (standard for certain colour spaces): (0.2126*R + 0.7152*G + 0.0722*B) source

• Luminance (perceived option 1): (0.299*R + 0.587*G + 0.114*B) source

• Luminance (perceived option 2, slower to calculate): sqrt( 0.241*R^2 + 0.691*G^2 + 0.068*B^2 ) → sqrt( 0.299*R^2 + 0.587*G^2 + 0.114*B^2 ) (thanks to @MatthewHerbst) source

^{_{[Edit: added examples using named css colors sorted with each method.]}}

I think what you are looking for is the RGB -> Luma conversion formula.

Photometric/digital ITU BT.709:

Y = 0.2126 R + 0.7152 G + 0.0722 B

Digital ITU BT.601 (gives more weight to the R and B components):

Y = 0.299 R + 0.587 G + 0.114 B

If you are willing to trade accuracy for perfomance, there are two approximation formulas for this one:

Y = 0.33 R + 0.5 G + 0.16 B

Y = 0.375 R + 0.5 G + 0.125 B

These can be calculated quickly as

Y = (R+R+B+G+G+G)/6

Y = (R+R+R+B+G+G+G+G)>>3

The "Accepted" Answer is Incorrect and Incomplete

The only answers that are accurate are the @jive-dadson and @EddingtonsMonkey answers, and in support @nils-pipenbrinck. The other answers (including the accepted) are linking to or citing sources that are either wrong, irrelevant, obsolete, or broken.

Briefly:

• sRGB must be LINEARIZED before applying the coefficients.
• Luminance (L or Y) is linear as is light.
• Perceived lightness (L*) is nonlinear as is human perception.
• HSV and HSL are not even remotely accurate in terms of perception.
• The IEC standard for sRGB specifies a threshold of 0.04045 it is NOT 0.03928 (that was from an obsolete early draft).
• To be useful (i.e. relative to perception), Euclidian distances require a perceptually uniform Cartesian vector space such as CIELAB. sRGB is not one.

What follows is a correct and complete answer:

Because this thread appears highly in search engines, I am adding this answer to clarify the various misconceptions on the subject.

Luminance is a linear measure of light, spectrally weighted for normal vision but not adjusted for the non-linear perception of lightness. It can be a relative measure, Y as in CIEXYZ, or as L, an absolute measure in cd/m² (not to be confused with L*).

Perceived lightness is used by some vision models such as CIELAB, here L* (Lstar) is a value of perceptual lightness, and is non-linear to approximate the human vision non-linear response curve. (That is, linear to perception but therefore non linear to light).

Brightness is a perceptual attribute, it does not have a "physical" measure. However some color appearance models do have a value, usualy denoted as "Q" for perceived brightness, which is different than perceived lightness.

Luma (Y´ prime) is a gamma encoded, weighted signal used in some video encodings (Y´I´Q´). It is not to be confused with linear luminance.

Gamma or transfer curve (TRC) is a curve that is often similar to the perceptual curve, and is commonly applied to image data for storage or broadcast to reduce perceived noise and/or improve data utilization (and related reasons).

To determine perceived lightness, first convert gamma encoded R´G´B´ image values to linear luminance (L or Y ) and then to non-linear perceived lightness (L*)

TO FIND LUMINANCE:

...Because apparently it was lost somewhere...

Step One:

Convert all sRGB 8 bit integer values to decimal 0.0-1.0

  vR = sR / 255;
  vG = sG / 255;
  vB = sB / 255;

Step Two:

Convert a gamma encoded RGB to a linear value. sRGB (computer standard) for instance requires a power curve of approximately V^2.2, though the "accurate" transform is:

Where V´ is the gamma-encoded R, G, or B channel of sRGB.
Pseudocode:

function sRGBtoLin(colorChannel) {
        // Send this function a decimal sRGB gamma encoded color value
        // between 0.0 and 1.0, and it returns a linearized value.

    if ( colorChannel <= 0.04045 ) {
            return colorChannel / 12.92;
        } else {
            return pow((( colorChannel + 0.055)/1.055),2.4);
        }
    }

Step Three:

To find Luminance (Y) apply the standard coefficients for sRGB:

Pseudocode using above functions:

Y = (0.2126 * sRGBtoLin(vR) + 0.7152 * sRGBtoLin(vG) + 0.0722 * sRGBtoLin(vB))

TO FIND PERCEIVED LIGHTNESS:

Step Four:

Take luminance Y from above, and transform to L*

Pseudocode:

function YtoLstar(Y) {
        // Send this function a luminance value between 0.0 and 1.0,
        // and it returns L* which is "perceptual lightness"

    if ( Y <= (216/24389)) {       // The CIE standard states 0.008856 but 216/24389 is the intent for 0.008856451679036
            return Y * (24389/27);  // The CIE standard states 903.3, but 24389/27 is the intent, making 903.296296296296296
        } else {
            return pow(Y,(1/3)) * 116 - 16;
        }
    }

L* is a value from 0 (black) to 100 (white) where 50 is the perceptual "middle grey". L* = 50 is the equivalent of Y = 18.4, or in other words an 18% grey card, representing the middle of a photographic exposure (Ansel Adams zone V).

References:

IEC 61966-2-1:1999 Standard
Wikipedia sRGB
Wikipedia CIELAB
Wikipedia CIEXYZ
Charles Poynton's Gamma FAQ

To add what all the others said:

All these equations work kinda well in practice, but if you need to be very precise you have to first convert the color to linear color space (apply inverse image-gamma), do the weight average of the primary colors and - if you want to display the color - take the luminance back into the monitor gamma.

The luminance difference between ingnoring gamma and doing proper gamma is up to 20% in the dark grays.

I was solving a similar task today in javascript. I've settled on this getPerceivedLightness(rgb) function for a HEX RGB color. It deals with Helmholtz-Kohlrausch effect via Fairchild and Perrotta formula for luminance correction.

/**
 * Converts RGB color to CIE 1931 XYZ color space.
 * https://www.image-engineering.de/library/technotes/958-how-to-convert-between-srgb-and-ciexyz
 * @param  {string} hex
 * @return {number[]}
 */
export function rgbToXyz(hex) {
    const [r, g, b] = hexToRgb(hex).map(_ => _ / 255).map(sRGBtoLinearRGB)
    const X =  0.4124 * r + 0.3576 * g + 0.1805 * b
    const Y =  0.2126 * r + 0.7152 * g + 0.0722 * b
    const Z =  0.0193 * r + 0.1192 * g + 0.9505 * b
    // For some reason, X, Y and Z are multiplied by 100.
    return [X, Y, Z].map(_ => _ * 100)
}

/**
 * Undoes gamma-correction from an RGB-encoded color.
 * https://en.wikipedia.org/wiki/SRGB#Specification_of_the_transformation
 * https://stackoverflow.com/questions/596216/formula-to-determine-brightness-of-rgb-color
 * @param  {number}
 * @return {number}
 */
function sRGBtoLinearRGB(color) {
    // Send this function a decimal sRGB gamma encoded color value
    // between 0.0 and 1.0, and it returns a linearized value.
    if (color <= 0.04045) {
        return color / 12.92
    } else {
        return Math.pow((color + 0.055) / 1.055, 2.4)
    }
}

/**
 * Converts hex color to RGB.
 * https://stackoverflow.com/questions/5623838/rgb-to-hex-and-hex-to-rgb
 * @param  {string} hex
 * @return {number[]} [rgb]
 */
function hexToRgb(hex) {
    const match = /^#?([a-f\d]{2})([a-f\d]{2})([a-f\d]{2})$/i.exec(hex)
    if (match) {
        match.shift()
        return match.map(_ => parseInt(_, 16))
    }
}

/**
 * Converts CIE 1931 XYZ colors to CIE L*a*b*.
 * The conversion formula comes from <http://www.easyrgb.com/en/math.php>.
 * https://github.com/cangoektas/xyz-to-lab/blob/master/src/index.js
 * @param   {number[]} color The CIE 1931 XYZ color to convert which refers to
 *                           the D65/2° standard illuminant.
 * @returns {number[]}       The color in the CIE L*a*b* color space.
 */
// X, Y, Z of a "D65" light source.
// "D65" is a standard 6500K Daylight light source.
// https://en.wikipedia.org/wiki/Illuminant_D65
const D65 = [95.047, 100, 108.883]
export function xyzToLab([x, y, z]) {
  [x, y, z] = [x, y, z].map((v, i) => {
    v = v / D65[i]
    return v > 0.008856 ? Math.pow(v, 1 / 3) : v * 7.787 + 16 / 116
  })
  const l = 116 * y - 16
  const a = 500 * (x - y)
  const b = 200 * (y - z)
  return [l, a, b]
}

/**
 * Converts Lab color space to Luminance-Chroma-Hue color space.
 * http://www.brucelindbloom.com/index.html?Eqn_Lab_to_LCH.html
 * @param  {number[]}
 * @return {number[]}
 */
export function labToLch([l, a, b]) {
    const c = Math.sqrt(a * a + b * b)
    const h = abToHue(a, b)
    return [l, c, h]
}

/**
 * Converts a and b of Lab color space to Hue of LCH color space.
 * https://stackoverflow.com/questions/53733379/conversion-of-cielab-to-cielchab-not-yielding-correct-result
 * @param  {number} a
 * @param  {number} b
 * @return {number}
 */
function abToHue(a, b) {
    if (a >= 0 && b === 0) {
        return 0
    }
    if (a < 0 && b === 0) {
        return 180
    }
    if (a === 0 && b > 0) {
        return 90
    }
    if (a === 0 && b < 0) {
        return 270
    }
    let xBias
    if (a > 0 && b > 0) {
        xBias = 0
    } else if (a < 0) {
        xBias = 180
    } else if (a > 0 && b < 0) {
        xBias = 360
    }
    return radiansToDegrees(Math.atan(b / a)) + xBias
}

function radiansToDegrees(radians) {
    return radians * (180 / Math.PI)
}

function degreesToRadians(degrees) {
    return degrees * Math.PI / 180
}

/**
 * Saturated colors appear brighter to human eye.
 * That's called Helmholtz-Kohlrausch effect.
 * Fairchild and Pirrotta came up with a formula to
 * calculate a correction for that effect.
 * "Color Quality of Semiconductor and Conventional Light Sources":
 * https://books.google.ru/books?id=ptDJDQAAQBAJ&pg=PA45&lpg=PA45&dq=fairchild+pirrotta+correction&source=bl&ots=7gXR2MGJs7&sig=ACfU3U3uIHo0ZUdZB_Cz9F9NldKzBix0oQ&hl=ru&sa=X&ved=2ahUKEwi47LGivOvmAhUHEpoKHU_ICkIQ6AEwAXoECAkQAQ#v=onepage&q=fairchild%20pirrotta%20correction&f=false
 * @return {number}
 */
function getLightnessUsingFairchildPirrottaCorrection([l, c, h]) {
    const l_ = 2.5 - 0.025 * l
    const g = 0.116 * Math.abs(Math.sin(degreesToRadians((h - 90) / 2))) + 0.085
    return l + l_ * g * c
}

export function getPerceivedLightness(hex) {
    return getLightnessUsingFairchildPirrottaCorrection(labToLch(xyzToLab(rgbToXyz(hex))))
}

Consider this an addendum to Myndex's excellent answer. As he (and others) explain, the algorithms for calculating the relative luminance (and the perceptual lightness) of an RGB colour are designed to work with linear RGB values. You can't just apply them to raw sRGB values and hope to get the same results.

Well that all sounds great in theory, but I really needed to see the evidence for myself, so, inspired by Petr Hurtak's colour gradients, I went ahead and made my own. They illustrate the two most common algorithms (ITU-R Recommendation BT.601 and BT.709), and clearly demonstrate why you should do your calculations with linear values (not gamma-corrected ones).

Firstly, here are the results from the older ITU BT.601 algorithm. The one on the left uses raw sRGB values. The one on the right uses linear values.

ITU-R BT.601 colour luminance gradients

0.299 R + 0.587 G + 0.114 B

At this resolution, the left one actually looks surprisingly good! But if you look closely, you can see a few issues. At a higher resolution, unwanted artefacts are more obvious:

The linear one doesn't suffer from these, but there's quite a lot of noise there. Let's compare it to ITU-R Recommendation BT.709…

ITU-R BT.709 colour luminance gradients

0.2126 R + 0.7152 G + 0.0722 B

Oh boy. Clearly not intended to be used with raw sRGB values! And yet, that's exactly what most people do!

At high-res, you can really see how effective this algorithm is when using linear values. It doesn't have nearly as much noise as the earlier one. While none of these algorithms are perfect, this one is about as good as it gets.

RGB Luminance value = 0.3 R + 0.59 G + 0.11 B

http://www.scantips.com/lumin.html

If you're looking for how close to white the color is you can use Euclidean Distance from (255, 255, 255)

I think RGB color space is perceptively non-uniform with respect to the L2 euclidian distance. Uniform spaces include CIE LAB and LUV.

The HSV colorspace should do the trick, see the wikipedia article depending on the language you're working in you may get a library conversion .

H is hue which is a numerical value for the color (i.e. red, green...)

S is the saturation of the color, i.e. how 'intense' it is

V is the 'brightness' of the color.

The 'V' of HSV is probably what you're looking for. MATLAB has an rgb2hsv function and the previously cited wikipedia article is full of pseudocode. If an RGB2HSV conversion is not feasible, a less accurate model would be the grayscale version of the image.

As mentioned by @Nils Pipenbrinck:

All these equations work kinda well in practice, but if you need to be very precise you have to [do some extra gamma stuff]. The luminance difference between ignoring gamma and doing proper gamma is up to 20% in the dark grays.

Here's a fully self-contained JavaScript function that does the "extra" stuff to get that extra accuracy. It's based on Jive Dadson's C++ answer to this same question.

// Returns perceived brightness (0-1) of the given 0-255 RGB values
// Based on this C++ implementation: https://stackoverflow.com/a/13558570/11950764
function rgbBrightness(r, g, b) {
  let v = 0;
  v += 0.212655 * ((r/255) <= 0.04045 ? (r/255)/12.92 : Math.pow(((r/255)+0.055)/1.055, 2.4));
  v += 0.715158 * ((g/255) <= 0.04045 ? (g/255)/12.92 : Math.pow(((g/255)+0.055)/1.055, 2.4));
  v += 0.072187 * ((b/255) <= 0.04045 ? (b/255)/12.92 : Math.pow(((b/255)+0.055)/1.055, 2.4));
  return v <= 0.0031308 ? v*12.92 : 1.055 * Math.pow(v,1.0/2.4) - 0.055;
}

Please define brightness. If you're looking for how close to white the color is you can use Euclidean Distance from (255, 255, 255)