Everyone is welcome to contribute to Wikipedia, regardless of whether they choose to register a username or not. However, you must have a registered username in order to create new articles.
There are two main reasons people are encouraged to register:
The free account gives you many benefits, including more editing options and user preferences. One handy feature is the watchlist, which makes it easier for you to track changes to pages you are interested in. Another is the ability to move or rename a page. (Do not move pages simply through copying and pasting as the edit history is not preserved that way. Instead, ask at the Village pump for a registered user's help.)
An unregistered user is identified by his or her machine's IP address, which is used as their public identifier when making contributions (and signing comments on talk pages). Your computer's IP address can sometimes be used to find information about you, so registering increases your privacy by hiding it. Also, because IP addresses often change, unregistered users sometimes find it more difficult to gain respect and recognition from other editors through building a good track record. It can also be more difficult to carry on conversations with such non-registered users since they lack a unique talk page. For these reasons, many Wikipedians consider registering to indicate a greater willingness by you to stand behind your edits, and your comments on discussion pages. This means your contributions to articles and discussion pages will generally be given more weight.
Finally, only registered users are allowed to become administrators (also known as sysops). If you register, do not forget your password or your user name. If you're prone to forgetting these, make sure you enter your email address and verify it when you sign up so you can have a new password sent to you if you forget your current one.
To register you can click the Sign in link at the top right corner of any page. You won't be able to (easily) change your username once you choose it, so reading the username policy before creating a username is highly recommended.