One of the way to do it is to add left padding to the field and add background image for the field.
See example: http://jsfiddle.net/hYAEQ/
It's not exact way twitter.com do it, they used absolute position element above search field because they have all images in the single sprite, and can't easily use them as backgrounds, but it should do.
I used inline image for a background to make it easier to post it to jsfiddle, but feel free to use normal links to images here.
EDIT: The way to do it using bootstrap sprite and additional container for icon
http://jsfiddle.net/hYAEQ/2/
EDIT 2:
Fix for white bootstrap theme: http://jsfiddle.net/hYAEQ/273/
EDIT 3:
If you are using navbar-inverse (black navbar) you will want this minor tweak: http://jsfiddle.net/hYAEQ/410/
.navbar-search .search-query {
padding-left: 29px !important;
}