Okay, having seen a few artists here get burned recently over issues of copyright and so on, I want to lay out a few simple guidelines. This is horribly over-simplified, but should make certain things clear.

Copyright is on a piece of work, literary, musical, artistic, etc. This is why Sony can copyright a disc of Beethoven's 5th; they haven't copyrighted the symphony, they've copyrighted the particular recording they're selling. So an artist can draw someone else's character and guess what? The artist holds the copyright on the drawing, not the other person. Most places, copyright automatically exists from the moment the work is created, though if you have to go to court over a copyright issue, registering the copyright will help you prove it. Copyright expires at different times depending on where you live, but as long as it's in force, the creator of that work has it.

Trademark is different; trademark protects symbols and logos used in commerce, hence 'trade'. Disney trademarks all its characters because it uses them as representations in business, including selling stuff with the characters on it. Also, trademark has to be defended if infringed; if a business does not defend (i.e., litigate) infringements on its trademark, a court may rule the company abandoned that trademark. That's why Disney goes after fanart of their characters so vigourously.

Pubic domain is stuff that used to be copyrighted and the copyright has expired, or the creator may decide to make it public domain right from the get-go. This doesn't mean someone else can come along and slap a copyright or trademark on it, though; once it's public domain, it stays public domain.

So, commissioners, when you commission a piece of art from an artist, unless you specifically arrange to purchase the copyright, it remains with the artist. Some artists offer 'private commissions', meaning they will not display that piece anywhere else, but they haven't given up the copyright. Some commissioners have gone so far as to remove the artist's signature/copyright and substitute their own, under the mistaken belief that since it's their character, it's their copyright, but that's clear and blatant copyright infringement, and in fact is grounds for court action.

Myself, I prefer to make sure I get clear statements from an artist regarding what I may and may not do with the art I get from them - can I post it on other sites, can I crop it to make icons, can I colour it, can I add a thing like "Rhiannon is the property of GingerM" to it. On the flip side, the stuff I've created and put up is very much copyright to me, no matter that it may have other characters in it, or even if it has none of my characters in it. I have a couple of things in mind for the future which I may make public domain, in which case I'll make it clear they are such. But if someone else comes along and tries to claim they can take ownership of them, they'll find out just how wrong they are.

ETA: Another common misconception is the idea that anything goes on the internet. Yes, copyright does exist on the internet, so ripping art and re-posting to uchan or e621 or whatever is copyright infringement. This is one of those situations, by the way, where strategically placed watermarks are the original artist's best friend if it becomes necessary to prove the origin of the work to these sites.