UPDATE: Seeing that many liked some of the additional tricks I've mentioned here, be sure to check the "Other (cool) boundaries section" at the bottom
I feel that none of the answers is complete:
In general, you usually start a delete operation using d<motion>, and seldom using x.
Note: When N is not specified, vim behaves as if N=1 (deletes a single char)
Discrete characters:
<N>x - Delete N chars to the right
d<N><left-arrow> - Delete N chars to the left
d<N><right-arrow> - Delete N chars to the right
Word boundaries:
Note: The 1st preceding/succeeding word is the one under the cursor
d<N>b - Delete from the beginning of the preceding N-th word to the current position
d<N>e - Delete from current position to the end of the succeeding N-th word
d<N>w - Same as d<N>e but including trailing whitespace
diw - Delete the entire word under the cursor
daw - Same as diw but including trailing whitespace
Line boundaries:
d0 - Delete from the beginning of the line to the current position
d^ - Delete from the first non-whitespace char to the current position
d$ - Delete from the current position to end of line
Other (cooler) boundaries:
These are tricks even veteran vim users usually don't know of, but are probably some of the most powerful tools in vim.
di( or di) - If you are located anywhere inside a parenthesis scope, i.e. (int a, int b, int c), deletes everything between the parenthesis, and leaves (). Very useful when changing function prototypes, conditions, loops, etc.
di{ or di} - Same as the one above, but deletes everything inside a curly brace scope. Very useful for changing a method implementation easily.
di[ or di] - You get the point, right? ;)
dit - Delete contents of a tag. For example, given <a href="..."><img src="smiley.gif"></a>, if you place you cursor within the <a...> tag and press dit, it will delete the entire <img...> part (Thanks @Eric for mentioning that).
Other important hint[s]:
Just in case you didn't know that, every sequence that starts with d can be replaced with c to delete and then go straight into insert mode!