Many furs choose to use an alias rather than their real name (or some other screen name) when posting to the group. There are many reasons to do this, for instance you may find it easier to relate to your furriness using an assumed name, or want to express something of your fur side in your choice of name. You may already by well known in other furry circles, such as MUCKs, by a character name. Perhaps one of the biggest reasons though is anonymity.
When you post to a newsgroup, your posts are archived by service such as Google Groups. These services allow you to easily find newsgroup articles of interest, but they also make it easy for others to track down posts you have made. If you are concerned about others (family, friends, employers) being able to read what you post to the group, one of the simplest things to do is use an assumed name. This makes it significantly more difficult to find your posts if someone already knows another address you use. In order for this to work you must keep the two names separate though, use one only for newsgroups, and the other for friends and work. It only takes one slip-up and the whole thing can be unravelled.
Conversely, you may not want to hide your furriness and chose to use your own name on the group. Neither option is right or wrong, and you'll meet furs doing both on the group.
Some furs who have a spiritual connection to their fur side know the name that their spirit uses. This can be revealed by dreams or other methods. You may want to think twice about using your spirit name to post to the group, given its very personal nature.
One of the biggest problems in choosing a name is finding one that isn't already in use. It can be a problem if you de-lurk and lots of people you don't know start saying "Welcome back, we haven't seen you in years!", worse still if you pick a name previously used by someone unpleasant.
Perhaps understandably, if someone has been using a name on-line for many years they feel (rightly or wrongly) that they own the name. If despite all precautions you find a conflict with another fur it is best to choose a new name.
You can check the names of all the posters who have used alt.lifestyle.furry in the posting statistics part of this site. The last summary file linked from that page lists all the posters to the group. This file may take a while to download as at the time of writing it listed over 3000 names and was a shade over 200k. Use your browser's search for text feature to check if a name has been used.