And on cue...
Well, other than combustion and pneumatics, other common launching methods include, stored strain energy (like in an onager, or crossbow), gravitational energy (like in a trebuchet), solid propellants (not safe, legal, or quiet, don't mess with them) or electric power.
Electric power can be split into two subsections:
- Electromagnetic
- Electrothermal
Electromagnetic can be spilt down further into reluctance, and inductance.
In simple terms, reluctance is using an attractive force, inductance a repulsive one.
Railguns and Thompson's coils are always inductive, coilguns can be either. When people normally talk about a coilgun, they're referring to a reluctant coilgun, using magnetic fields to draw in a steel armature (or projectile), but they can also be inductive, producing high currents within the projectile itself with sheer electromagnetic brute force (which creates a magnetic field), and then repelling that with the same field that created the currents in the projectile. That takes a lot more power, but it's also I believe how the world's fastest coilgun currently works.
Electrothermal cannons use electric power to create heat to expand gases to force a projectile with pressure. You can either heat the air with a vast spark, but that's obviously inefficient.
A common variant is the atomisation of water with a high power electric impulse. Basically, if you apply enough power to water quickly enough, it turns to steam almost instantly, and you can create huge pressures with enough application of power.
That brings me on to one last type of cannon. The steam cannon, which flash boils water in a fraction of a second, again, producing vast pressures.
That lists all the main ways I can think of to launch something. There are other ones, but those are the variants that a home hobbyist might be able to achieve.
For more info on Electric based acceleration, I'd advise taking a look around:
http://www.powerlabs.org/emguns.htm
Also take a look at the atomisation of water section on the high voltage section of that site.