You just got yourself on some government list for having an interest in guns, MR! Surely this must mean you are unstable and could freak-out at any moment.
<< Pulling the trigger causes the firing pin to strike the primer of the cartrige, the primer ignites causing the gunpowder to ignite causing a high pressure system inside the case which forces the bullet into and out of the barrel. >>
That's it in a nut-shell!
There was a cool firearm tutorial on the internet showing how a handgun worked, but the link is dead now. Different firearms have slightly different mechanisms, but the basic concepts are all the same, except for case-less ammunition, which is really cool.
Primers are made of small amounts of lead-azide, fulminate of mercury, lead styphenate, potassium chlorate, or other compounds which are friction-sensitive explosives. When struck with sufficient force, these small primary explosives ignite the main propellent (gun powder). Gun powder burns vigorously and the gases released from combustion rapidly expand inside the case and push the bullet down and out of the barrel at anywhere from several hundred feet per second to a few thousand feet per second.
Not much more to say about that.