The AR-15 is a trademarked name owned by Colt which they use to designate their semi-automatic line of civilian rifles. They were originally designed to fire the 5.56mm NATO round, but can also safely discharge the civilian .223Rem in most instances. Today there are a variety of interchangeable upper halves available that are offered in many different calibers such as 7.62X39mm, .204 Ruger, 6.8mm SPC, 6.5 Grendel, etc...
The AR-10 is the AR-15's bigger brother and is usually chambered in the 7.62mm NATO or .308Winchester round. Armalite, DPMS and several other manufacturers offer the semi-automatic AR-10 for sale to civilians. While they share some parts in common with the AR-15, not all are interchangeable.
Today, the terms AR-15 & AR-10 are commonly used to refer to all rifles of this type, regardless of the manufacturer. Due to trademark restrictions, each maker is required to come up with their own "unique" trademarked designation for their rifle; usually some variation of 15 or 10 is found in the name.
While you haven't asked about it specifically, the M4 Carbine is another variant of the AR-15 that usually sports a 14.5" or 16" (All AR/M4 rifles in Canada are classified as Restricted so barrel length really doesn't factor into the equation. In the US, however, they have a legal barrel length restriction of 16" for civilian ownership. Anything shorter than that must be registered with the BATF as an SBR or short-barrel rifle and a tax stamp that costs $200 acquired in order to legally posses an assembled SBR) barrel with a special step-cut countour that allows for an under barrel grenade launcher to be used. Like the AR-15, the term M4 is used so often today, it can be used to refer to any M4 clone or "M4gery" made by any of today's manufacturers.
In all of the above cases, the rifles in question are semi-auto only and aren't technically "assault rifles."
The M-16, on the other hand, is a select fire military issued small arm that is capable of firing in semi-auto, as well as automatic or three round burst depending on the variant.
Colt remains the pre-eminent AR manufacturer and their product remains the most sought after and collectible. However, Colt is reknowned for their anti-civilian attitude and the company really isn't very regarded in terms of customer service for civilians. This has contributed to something of a decreased availability of Colt products and has driven prices up accordingly, especially here in Canada. While generally well made, there are other manufacturers today turning out products that are every bit the equal of the Colt, just not with the prestige and cache the prancing pony brings with it.
Probably the best of the manufactures after Colt is LMT. They make an extremely high quality product, as does Noveske.
Other manufactures like CMMG, STAG/CMT, turn out excellent AR rifles. Other well known manufacturers include Armalite, Bushmaster, Rock River Arms, DPMS, etc...
