Mage knights (also often referred to as spellswords) are magicians or knights who decide they want to walk a mile in the others' shoes. The generally quickly realize that their previous experience and equipment plus their new experience and equipment make them fuckawesome at pretty much everything. They can blow shit up with magic, kill stuff up with swords, and generally be a badass motherfucker.
Nearly all settings have some form of rule saying that magic doesn't work while the caster is wearing armor. Although being a mage knight/spellsword generally lessens this effect, don't expect to be slinging meteors around while dueling with the enemy champion. Really, instead of an even half-and-half of wizard and warrior, they usually end up being fighters with some cool little tricks or mages that can use swords when times are desperate.
According to the Law of Cliche, all mage knights must have some sort of stupid crystal sword which focuses their magic or some shit. They also love to ride dragons. Although lesser offenders than professional dragon riders (yep, that's a real prestige class), it's still retarded, because how exactly are you supposed to use that extremely rare, extremely badass sword while on the back of a beast so large you barely reach up to its ankles?
Mage Knight is also the name of a commercial board game that can be played solo or with up to 4 players.