AR15 is actually a Colt Trademark more than anything else, the trademark was purchased from Armalite back in the late 50s if I remember correct and for a short time Colt continued to show the Armalite heritage with the "Armalite AR15" name rolled into the magazine well along with the Colt name. They eventually stopped though and held onto the AR15 rights with a pretty tight fist. They did however drop the AR15 name in recent years because of the laws that the other members have mentioned, now you've got various goofy model numbers and such.
Ever wonder why DPMS referrs to their rifles as "A15s" and make wierd names like "Hartford Horsey Logo" for Colt crap? It's because Colt has let loose the lawyers in the past over the use of the trademarked names.
California was the first to ban firearms by name back around 1989 and as a result a lot of companies immediately renamed their firearms to make them California legal again, reason for "sporters" and the such.
Getting generic the term "AR15" pretty much refers to all the same thing, a civilian semiauto version of the M16. Bushmaster XM15s, Armalite M15s, DPMS A15s, Essential Arms J15s, it's all pretty much the same stuff/same thing. Colt has seen fit to start using non-standard pins and things though on their newer rifles to try and stop people from using a set of M16 fire control parts in it, if you're looking for an AR15 that is about as close to Mil-Spec as you can possibly get than by all means go with a Bushmaster rifle.