So, I'm going to guess finish typing means you just stop for a while, say 5 seconds. So with that in mind, lets start a timer when the user releases a key and clear it when they press one. I decided the input in question will be #myInput.
Making a few assumptions...
//setup before functions
var typingTimer; //timer identifier
var doneTypingInterval = 5000; //time in ms, 5 second for example
//on keyup, start the countdown
$('#myInput').keyup(function(){
clearTimeout(typingTimer);
typingTimer = setTimeout(doneTyping, doneTypingInterval);
});
//on keydown, clear the countdown
$('#myInput').keydown(function(){
clearTimeout(typingTimer);
});
//user is "finished typing," do something
function doneTyping () {
//do something
}