While a bit political, [this Vox article does a good job explaining the history of Captain Marvel](https://www.vox.com/2014/11/5/7138099/captain-marvel-background-history-movie).

[Captain Mar-Vell](http://marvel.wikia.com/wiki/Mar-Vell_(Earth-616)) (people generally called him Marvel) was a [Kree](http://marvel.wikia.com/wiki/Kree). That's this guy:

[![Captain Mar-Vell][1]][1]

[Carol Danvers](http://marvel.wikia.com/wiki/Carol_Danvers_(Earth-616)) was originally someone who worked with Mar-Vell ([Marvel Superheroes #13](http://marvel.wikia.com/wiki/Marvel_Super-Heroes_Vol_1_13)). She even had a romantic relationship with the Kree version. Later, she had a run-in with a Kree device that made her half-Kree. Once she realized she had powers like Mar-Vell, she took the name herself (the story there was Mar-Vell left Earth).

[![Carol Danvers][2]][2]

Generally speaking, when people refer to Captain Marvel, they mean Carol Danvers, since she's held the title longer (the original was canceled in the 70s).

 [1]: https://i.sstatic.net/f6Hrr.jpg
 [2]: https://i.sstatic.net/4rKv4.jpg