The big issue is ‘fragmentation’ or breaking. DNA is a long molecule – for example, each chromosome is one long double helix of DNA and it can be 50 million or 100 million ‘bases’ (ACG or T) long. That is enormous! So when we try to purify the DNA (to get rid of stuff that isn’t DNA), then usually it will break into pieces. Also as soon as an animal or plant dies, the DNA starts to break into pieces naturally. It is very fragile. In a good quality DNA sample, the broken DNA pieces could be 100 thousand bases long (=100 kb) or more, and we can be OK with that. But in a bad quality DNA sample, the pieces might have got chopped up to just a few thousand bases – then it won’t sequence very well.
One of the important things in trying to get a “good” DNA sample is how you store that sample before you take the DNA out of the sample. As Peter says, DNA will start to break down naturally, so it is important to store the sample as carefully as possible (often by freezing at very low temperatures), to stop the DNA from breaking up. We are then very careful throughout all of the various lab teams to keep the sample at those low temperatures until we are ready to use the DNA.