How do mutations work in general, and in relation to infectious disease (bacteria and viruses)?

When replicating a genome, mistakes may happen. Wrong bases may be added, bases may be deleted etc.
Usually, the cell has a machinery to fix the mistakes, but sometimes the mistakes go unnoticed and that is when you get a mutation.

In bacteria and viruses, mutations happens more frequently than in humans. This is mostly due to two different things:
Bacteria can absorb DNA from its environment, and implement it into their own genome.
Viruses often have very small genomes, so they do not have the machinery to repair the mistakes that gets made.

This means that bacteria and viruses have frequent mutations, which makes them very resilient. This is the reason why you never get immune to the flu, because the viruses mutate and come back a little bit different.