Episode missions are PvE "storyline" content. They can be completed solo or with a group.
Exploration missions are dynamically generated missions in certain exploration areas. They can be completed solo or with a group.
Sector Control missions ask you to fly around and fight in the battles you see appearing on the sector map. These are PvE encounters and up to 20 players can participate.
Fleet Actions are scripted PvE encounters with up to 20 players participating. There will usually be three or four objectives that must be completed by the players and then a final boss (with friends). At the end of the fleet action, players are ranked based on how well they did, and receive rewards based on that. This is in addition to any mission rewards you get for completing the Fleet Action. You can only get the mission for a fleet action once, but you can do them as many times as you want outside of that.
Additional info: You must be lieutenant commander 1 to qualify for a new ship. Lieutenant 11 doesn't work.
Once you reach lieutenant commander, you must return to Earth Spacedock and talk with Admiral Quinn. You can not currently receive the free ship upgrade task from the Admiral via subspace communication. Just transwarp back to Earth Spacedock. He will then send you to a requisition officer that is located up the turbolift on the balcony overlooking the shipyard area. This officer can give you layman style details on what each of the ship classes do. There are also databases in the room which let you see the capabilities of each ship type.