To understand why we would use a DNA fingerprint, it's helpful to think why we use normal fingerprints to identify a person. Any two people you pick are going to have an enormous number of differences, pretty much everywhere if you look close enough. However, the fingerprint is one of the easiest parts of a person to look at and compare to another person.

With DNA it's a similar concept. Any two people (besides identical twins) will have different sets of DNA, called genomes. But, the genome is really, really large so for a long time it wasn't possible to look at 2 whole genomes and compare them to see if 2 people are different. The idea of DNA fingerprinting is that you only look at a very small part of the DNA that you expect to be different to determine if 2 DNA samples come from the same person.

The way this is done is that first a sample of DNA from a person is found. That sample is usually from human cells, blood, saliva or hair (with the root attached). The DNA is then cut up using molecules that cut DNA in a certain way. Then "probes" are added which light up when they bind a particular piece of DNA. In the case of DNA fingerprinting, these particular pieces of DNA are certain sequences that repeat over and over again, but the number of times they repeat is different in different people. This generates an image that looks like a bunch of bars, kind of like a barcode. In a real sense, the DNA fingerprint achieved is a barcode for a person in that it uniquely identifies people.

There are other ways to get a DNA fingerprint, but they all rely on looking at certain places in the genome that we expect to be different and comparing how these places look in different people. So the classic case is if a criminal leaves behind blood, they can do a DNA fingerprint on the blood of the criminal and the crime scene blood and see if the "bars" on the fingerprint are in the same places.