Others have answered how the redirects work but you should also know how they generate their tiny urls. You'll mistakenly hear that they create a hash of the URL in order to generate that unique code for the shortened URL. This is incorrect in most cases, they aren't using a hashing algorithm (where you could potentially have collisions).
Most of the popular URL shortening services simply take the ID in the database of the URL and then convert it to either Base 36 [a-z0-9] (case insensitive) or Base 62 (case sensitive).
A simplified example of a TinyURL Database Table:
ID URL VisitCount
1 www.google.com 26
2 www.stackoverflow.com 2048
3 www.reddit.com 64
...
20103 www.digg.com 201
20104 www.4chan.com 20
Web Frameworks that allow flexible routing make handling the incoming URL's really easy (Ruby, ASP.NET MVC, etc).
So, on your webserver you might have a route action that looks like (pseudo code):
Route: www.mytinyurl.com/{UrlID}
Route Action: RouteURL(UrlID);
Which routes any incoming request to your server that has any text after your domain www.mytinyurl.com to your associated method, RouteURL. It supplies the text that is passed in after the forward slash in your URL to that method.
So, lets say you requested: www.mytinyurl.com/fif
"fif" would then be passed to your method, RouteURL(String UrlID). RouteURL would then convert "fif" to its base10 equivalent, 20103, and a database request will be made to redirect to whatever URL is stored under the ID 20103 (in this case, www.digg.com). You would also increase the visit count for Digg by one before redirecting to the correct URL.
This is a really simplified example but you should be able to get the general idea.