How does the Hammas shoot explosives?

Via rockets. Like this one.

http://en.wikipedia.org/wiki/Qassam_rocket

Or others of similar design.

Car bombs have mostly gone down as a delivery system; the Israelis have made it pretty hard to get them across the border into anywhere they can cause damage.

Egypt

They usually fire 'rockets' they are called 'Qassam'. They have Russian parts but are usually 'homemade'. The warhead is like a hand grenade, they fill it with ball bearings, sharp metal bits to increase damage. It has a range of 3 to 10 miles and is unguided.

It's fired from a tripod backyard launcher or mortar, much like we fire fireworks for 4th July in US.

They also have some ex-Soviet 'Grad rockets' of the 1980's these need a support truck and launcher system have a 13mi range and a 122mm high explosive warhead (same as a tank shell), but are also unguided. You just point where you think you can do the most damage and shoot.

It's been widely reported Hamas has been trying to get actual Iranian missiles of 25mi to 800mi range with 2000-3000lb guided warheads.