Seriously, that's all there is to it. A Mystic Theurge is a PRC that improves your capabilities as both a Divine Spellcaster and as an Arcane Spellcaster. Great if you really can't choose between being a Cleric and a Wizard, or an Oracle and a Sorcerer, or whatever.
To get into this Prestige Class, you need to be able to cast at least 2nd level Arcane spells AND 2nd level Divine spells. You also need either 6 (for D&D) or 3 (for Pathfinder) ranks in BOTH Knowledge (Arcana) and Knowledge (Religion). In return, you get a ten level PrC where each level gives you spellcasting improvement in both your Arcane spellcasting class and your Divine spellcasting class.
Mystic Theurge is considered to be, when played as presented, not a very powerful class. You hand in three caster levels just to gain the ability to cast more spells. This means that at level 6, a Wizard or a Cleric would be only one level removed from 4th level spells while you can barely cast like a 3rd level in both. 10 levels later you can cast like a 13 level Wizard and Cleric, aka 7th level magic. That's only one level removed from 9th level magic for a level 16 Wizard or Cleric.
What is interesting to note is that neither Mystic Theurge nor the game itself defines just what arcane spellcasting classes and divine spellcasting classes are. With all the mess later in the game this line was rather muddled, putting all those fancy variety spellcasters in a sort of limbo regarding just what they are. This means that if a character is somehow both an arcane and a divine caster at the same time they can let both "+1 to arcane/divine spellcasting class" bonuses apply to the same class, effecively doubling how fast they level up their casting.
One way to do this is by ways of the Rainbow Servant. After going through the full 10 levels of this Prestige Class (note: you do really need the capstone for this) an Arcane caster is capable of learning and casting Cleric spells (just the basic ones, not from any domains) the same way they cast their regular spells. Regardless of how well it works for the various classes with access to the class (Bards and Sorcerers gain the least from it, while Wizards become even more flexible and Warmages becoming FUCKING HARDCORE) this now means that your character is both an Arcane and a Divine caster at the same time. Given that with some munchkinnery you can be eligible for this Prestige Class as early as level 2 (that is, your second level is Rainbow Servant) means that you can spend nine whole levels this way, effectively giving you a spellcaster level (and all the delicious goodness that comes with that) of 29 where it would normally be 20.
The Pathfinder version is a little meatier, in that it gains access to the Combined Spells class feature (which basically lets you give up one class's spellslot to prepare a spell from the other class of one level lower) and the Spell Synthesis class feature (which lets you cast an Arcane spell and a Divine spell simultaneously once per day).
In 5th edition, with the general approach of "class archetypes over prestige classes", the Mystic Theurge currently exists in four different options:
- The Arcana Cleric Domain provides Clerics with a selection of bonus Wizard spells and some "wizardy" Domain bonus features.
- The Favored Soul returned as a Sorcerous Origin called the "Divine Soul", which grants sorcerers some "angelic" traits and the ability to cast cleric spells.
- The Celestial Pact Warlock has some bonus Cleric spells focused on healing and can even heal with a touch.
- The Theurgy Tradition for Wizards grants the slow access to Cleric spells and the bonus class features of a chosen Cleric Domain. This is currently an "unofficial" option, as it exists only in Unearthed Arcana.