What determines whether an elimination reaction is E1 or E2?

The structure of the molecule and the strength of the base usually determines whether it will be E1 or E2

E2 reactions usually occur with secondary or tertiary substituted alkyl halides although a primary halide can also undergo the E2 reaction if a heavily substituted base is used. It also occurs with a strong base.

E1 reactions usually occur with tertiary alkyl halides, that is in the presence of a weak base.

You should review the E2 and E1 mechanism for further understanding of why structure of the alkyl halide and strength of the base affects whether the reaction will be E2 or E1.