Take any map — countries, US states, anything. Try to colour it so no two touching regions share a colour. You'll never need more than four colours.
The theorem
Proven in 1976 by Appel and Haken — the first major theorem proven by computer. They reduced the problem to checking 1,936 cases, then had a computer verify each.
Three colours are sometimes enough, sometimes not. Four is always enough. It's exact.