The basic objective is to infect (by touching) as many red blood cells as you can, and when you've infected 10, the body itself is infected.
As you infect more and more people, different medicines (aka enemies) are added into the mix. You can upgrade your bacteria, making it faster and upgrading your abilities.
It is also possible to spend Evopoints getting Adaptations, which enable new targets for infection. These mean you can infect more people at a time.