What is β-elimination, E2 and E1?

A $\beta$-elimination removes a leaving group Y from a carbon (the $\alpha$ carbon) and a proton from a neighbouring carbon (a $\beta$ carbon), forming a double bond between them. In the E2 mechanism a base takes the proton while Y leaves, in one elementary step; in the E1 mechanism Y first leaves, giving a carbocation, which then loses a $\beta$ proton.