The Elusive Neutrino

Billions of neutrinos are passing through your body this very second. They even cross the entire globe as if nothing is in their way. Their tiny mass is only few parts in million of the lightest atomic constituent, the electron. Moreover, they take part in almost no process. They cannot be seen, and typically travel at (almost) the speed of light. Then how in the world can scientists detect neutrinos? Why has nature ordered 3 different neutrino types? And what are they good for?