How do I increase the number of drones I can deploy?

Currently, I can only deploy three but some of the ships have bigger drone bays and I can't deploy them all. How do I increase the number I can deploy at one time? Is it skill based or something else (additional)?

There is an excellent guide available over at the EVE Wiki if you want more details, but in short there are three factors governing how many drones you can field:

1. Your Drones skill, which goes from 1 to 5, determines how many active drones you can keep track of;
2. Your ship's drone bandwitdh, measured in MBit/sec. Light drones take 5 MB/s, medium 10 MB/s, and heavy drones 25MB/s. The total amount of bandwidth your drones take up may not exceed the ship's bandwidth capability, so if your ship has only 25 MB/s bandwidth you're restricted to five small drones, two medium and three small, or one heavy drone.
3. Your ship's drone bay capacity, measured in cubic meters. As with bandwidth, you can't store more drones in your bay than you can fit in there. Any more go into your cargo hold. I suppose it's technically possible to load some in there, launch the drones you have, then crossload and launch more, but it'd only be an useful option in a vanishingly small cross section of ships and no help at all if you for some reason need to leave quickly. Actually, that's not possible.

TL;DR Until you get to capital ships (IE, carriers and motherships) there is no way you'll be able to control more than five drones at once. That said, with the right rigging and training, five tech2 heavy drones can do truly ridiculous amounts of damage. Ask your local Gallente ship dealer for a demonstration of the Ishtar.