Fission and Fusion Bombs

Fission = the splitting apart of atomic particles. When they break, they lose some mass (imagine this as the 'glue' that held them together) which is released as energy (as in E=MC2). In an atomic explosion, the pieces that originally split smash into other atoms, which then break and hit still more, etc, etc. This is the type of explosion that the bombs dropped on Hiroshima and Nagasaki utilized.

Fusion = the joining together of atomic particles. When these (smaller) particles join, a tiny portion of their mass is converted into energy (as in E=MC2). A weapon with this type of explosion is called a hydrogen bomb and can be significantly more powerful than a simple fission device.
Interestingly, the best way to 'smash' the pieces together tightly enough to fuse is by compressing them with an atomic explosion. Thus, a fusion device is first 'triggered' by a fission explosion - squeezing the core tightly enough together - which result in the fusion reaction beginning.