E1 reactions are unimolecular whereas E2 reactions are bimolecular. By extension,
#R_("E"1) = k["LG"^-]#, and
#R_("E"2) = k["LG"^-]["NUC"]#
where the notations stand for leaving group and nucleophile.
As a consequence of the preceding, E2 reactions usually proceed with a strong nucleophile (e.g. base), whereas E1 reactions are fine with a weaker nucleophile (e.g. base).
Mechanistically, E2 reactions are concerted (and occur faster), whereas E1 reactions are stepwise (and occur slower and at a higher energy cost, generally).
Due to E1's mechanistic behavior, carbocation rearrangements can occur in the intermediate, such that the positive charge is relocated on the most stable carbon. Factors influencing this include hyperconjugation and resonance. As a result of this, E1 usually won't occur on primary substrates.
Moreover, when competing with SN1, E1's rate is more sensitive and will dominate when supplied with heat.
If you'd like to see mechanisms, ask!