Different elements can have different isotopes-- the same element but with a different number of neutrons in the nucleus. U-235 and U-238 are both Uranium, but they are different isotopes of uranium.
Fission is the process where an unstable isotope's nucleus breaks apart. There are not many isotopes of any sort that undergo fission. U-235, U-233, P-239 are examples. When the nucleus of a fissionable isotope disintegrates, you will end up with atoms of completely different elements, free neutrons, and lots of energy (mass converted into energy by E=MC^2).
The fissionable isotopes I mentioned above are the ones which undergo fission upon neutron bombardment- as such the neutrons released from the fission can feed more fission in what is known as a chain reaction, which occurs when there is enough of the material present in a small area as is needed to sustain the reaction. This is known as critical mass. Thorium-232 is another fissionable isotope (its nucleus will break apart) but unlike the others, neutron bombardment will not cause the fission. It just happens over time.
Some non-fissionable isotopes (and it is not limited to the elements listed above), when bombarded by neutrons, will change. U-238, for example, when bombarded with neutrons, may accept a neutron to become U-239, which is itself an unstable isotope that is a beta emitter with a pretty short half life. When an atom emits a beta particle, it can be thought of as a neutron in the nucleus becoming a proton and an electron, with the electron being expelled (the beta particle). The new proton causes the atomic number of the atom to increase (atomic number merely being the number of protons in the nucleus-- which says what element it is), and as such what happens is that the U-239 becomes Pu-239.