Does IMDB provide an API?

IMDB themselves seem to distribute data, but only in text files:

http://www.imdb.com/interfaces

there are several APIs around this that you can Google. Screen scraping is explicitly forbidden. A official API seems to be in the works, but has been that for years already.

What about TMDb API ?

You can search by imdb_id with GET /find/{external_id}

https://developers.themoviedb.org/3/find/find-by-id

Yes, but not for free.

.....annual fees ranging from $15,000 to higher depending on the audience for the data and which data are being licensed.

URL :- http://www.imdb.com/licensing/

https://deanclatworthy.com/tools.html is an IMDB API but has been down due to abuse.

Here is a simple solution that fetches shows by name based on the query from Krinkle:

You can get around the same-origin policy by having your server fetch the URL instead of trying to fetch it directly with AJAX and you don't have to use JSONP to do it.

Javascript (jQuery):

function getShowOptionsFromName (name) {
    $.ajax({
        url: "ajax.php",
        method: "GET",
        data: {q: name},
        dataType: "json"
    }).done(function(data){
        console.log(data);
    });
}

PHP (in file ajax.php):

$q = urlencode($_GET["q"]);
echo file_get_contents("http://www.imdb.com/xml/find?json=1&nr=1&tt=on&q=$q");

Recently at SXSWi 2012, in their "Mashery Lounge", there was a booth for an IMDB-like API called from rovi. It's not a free API, but according to the sales guy I talked to they offer either a rev share or a flat fee for usage, depending on your budget. I haven't used it yet but it seems pretty cool.