A bomb is usually a casing of some sort, filled with explosive. The difference between a bomb and a shell is a bit vague, but a shell is usually fired from something like a gun, and is thus a projectile, where a bomb is dropped or placed. Of course, mortar bombs don't follow this plan, but it's a rule of thumb that generally works.
Bombs come in various sorts, including 'dumb' (with no guidance), laser guided (LGB) or 'smart', nuclear, armour piercing etc. When a daddy bomb and a mummy bomb cuddle in a special way, you get little Bomblets which live in a cluster bomb or shell.
The biggest bomb you can get away with dropping without starting WW3 or having Greenpeace sic your ass is the GBU-43/B MOAB (Massive Ordnance Air Blast or if you prefer Mother Of All Bombs). Read about how F off this bomb is here.