There's three versions. The F-35B is the one that can take off in a short distance and land vertically[0] and it has a big lift fan behind the cockpit. The F-35A and F-35C don't have the lift fan; the C has large wings and a reinforced tailhook[1] compared to the other versions.

[0] The F-35B _can_ take off vertically, but it can't do so with any reasonable weapons or fuel load.

[1] Many non-Navy planes have tailhooks to work with emergency arresting wire systems at Air Force bases, but those are for emergencies and are rarely used, whereas the Navy uses tailhooks all day every day

Only the B, and the B is only intended to do short takeoffs not vertical ones (it is possible to take off vertically but pointless, you can't do it with a combat load).

The C is meant to do carrier takeoffs and landings. Landing on a carrier the traditional way is more reliable than trying to land vertically every time.