Why are restriction enzymes important for DNA fingerprinting?
Since all organisms (from independent zygotes) possess unique DNA, the restriction enzymes will cut the DNA at different positions and different frequencies. This results in different numbers of "chunks" of varying lengths/sizes.
Restriction Fragment Length Polymorphisms (RFLP's) is the analysis of the fragments produced from a given restriction enzyme - the fragments are partially charged and will respond to electric fields.
The enzyme is important because the "fingerprint" produced is dependent on the different sized pieces of cut DNA taking different amounts of time to push through the mobile phase (typically an agarose gel if I'm not mistaken).