It's pretty simple:
- Equip the materia to the slots in your characters' current equipment.
- Fight monsters using the characters with that materia equipped.
From winning fights, you gain AP which will level the equipped materia. The equipment the materia is attached to can affect the amount of AP gained. Once a materia reaches its max level, you gain a new unleveled copy of that materia. Leveling all materia of a type can gain you the master materia which has the abilities of all materia of that type.
This site gives a fair rundown.
Materia grows by being equipped to someone who is in a battle, whether they use the materia or not. After each battle you get EXP points (which help your character's level increase) and AP points (which are like experience points for your materia, the more points your materia have the stronger they are and the more capabilities they have).
Be careful because some weapons do not permit materia growth, although most do. There are some weapons that double the AP points your materia receive and a few that even triple the AP points--check out the list of everyone's weapons in the equipment section for more details.
When your materia gather enough AP they become full or mastered (this is not the same as MASTER MATERIA). A mastered materia is as strong as it can possibly get, it won't gain any more AP but it will cause a brand new version of itself to be "born" in your materia pack and you can keep using the mastered one or you can start growing the newly born one. So, for example, even though there is only one Contain materia in the whole game, if you keep growing it you can have as many as you want.