URL shorteners take a long URL and create a short alias for it. One of the oldest ones, if not the oldest, is TinyURL. The user interface is simple: you take your long link, paste it into the box on the site and click the button. The long link above then becomes a much more manageable "http://tinyurl.com/cs92l4u". On the site you can, in theory, also choose a custom name for your URL, instead of the randomly assigned string "cs92l4u". However, TinyURL is so old and popular that most good short names are already taken. If you want to know where a TinyURL link leads to, you need to add "preview." in front of the domain name like this: http://preview.tinyurl.com/cs92l4u.
is.gd, pronounced "is good". You can choose to have it log statistics and it also creates a QR code for your URL, but there's no registration or login required. The example link shortened to http://is.gd/yitefe with the "lower case pronounceable" option selected. The preview page is the URL with a minus sign added, and from that page you can also access the statistics (if you chose them to be logged) and the QR code: http://is.gd/yitefe-. is.gd is also newer and less popular than TinyURL so you actually have a chance of getting some meaningful short name assigned to your link -- for example, the http://is.gd/bookofjobs address now leads to my post about the biography of Steve Jobs.
Tiny Arrows. The URLs created with it look really special: they actually have tiny arrows in them! Tiny Arrows never assigns a random string to your URL so you are forced to think of a name for your link, which is not bad. I shortened the example URL to this: http://➡.ws/senatesquare. What's also special about this shortener is that it doesn't take you to the target right away but actually shows a short countdown during which you see the actual URL of the page you are about to enter. You can also bypass this countdown by clicking a link or disable it altogether from ever appearing to you again.
mcaf.ee shortener by the computer security company McAfee. Because they are a reputable computer security company, they have a pretty good idea on which web sites are safe and which ones may try to infect you with viruses. So, the mcaf.ee links are safe to click. They also display a frame at the top of the page telling you security information about it. The example URL to Google Maps that I've been using in this post is a bad example in this case because it breaks out of the frame, but take the Steve Jobs post again: http://mcaf.ee/mgj7b. You'll see the "This site is Safe" text at the top of the page, along with a link to a detailed site report (which, in the case of this blog, tells you almost nothing, but anyway..).
ShadyURL is the opposite of mcaf.ee: it creates links that look really, really suspicious: something along the lines of http://5z8.info/best-russian-sites-for-bootleg-everything_r7c1lr_b00bs for example. It's more of a satire of an URL shortener than an actual URL shortener. :) And finally, if all those links were too short for you, then HugeURL is your service: it expands any URL into a ridiculously long one!
OK, we are almost in the end. There are many utility services also built around the concept of URL shorteners. For example, there are various browser plugins. For Chrome, the bitly plugin creates and copies short URLs to clipboard with one click. There are also services that show you the expanded links, possibly as tooltips over the shortened URLs. For Chrome, there's View Thru for that. For Firefox there are URL Shrink Easy and Long URL Please for the same purposes.
Finally, a word of warning: like any other website, a URL shortener site may stop working. At that point all of your short links will also stop working. This is called link rot. So keep your important links in the full form and only use shorteners if you figure that no damage will be done in case the link would stop working in a week or in a year. You can lower the risk of your links rotting by only using reputable shorteners, such as TinyURL and bitly, even though there are actually dozens of URL shorteners available.