Active skills are those that you can use during combat (if you have sufficient MP). Passive skills are always functioning.
You always have 5 available slots for active and 5 available slots for passive skills. If you have learned more than 5 skills in a category, you will have to choose which skills to have available (i.e. by assigning them to a slot) and which skills to keep in reserve. Skills can be switched into and out of slots at any time from the main interface, but cannot be changed during a combat.
Hover your mouse over a skill to learn about what it does.