The typical bomb consists of a large, lightly cased warhead1 fitted with some kind of fuse and, in most cases, some kind of tail section with stabilising fins and propellant.
The earliest mortar bombs were roughly spherical, cast iron balls filled with gunpowder and fitted with a burning fuse match. They were fired from muzzle loading, black powder mortars by a manual trigger (usually a cannon lock or flintlock).
The modern mortar bomb is cylindrical and has a finned tail (occasionally with pop-out over-calibre fins) which contains its propellant and firing cap - these are either fired when they hit the bottom of the mortar tube, or trigger fired, depending on design.
Other, mostly experimental designs have included overcalibre devices fired from the mortar by a rod inserted into the tube (a Japanese experiment from WW2) and self-obturating things that looked a lot like regular artillery shells (as found in a variety of rifled mortars such as the American 4.2" Chemical Mortar).
Some high technology bombs are designed to make a shaped charge round against the (usually weaker) top faces of armoured fighting vehicles, but these are not yet in general circulation.