An E1 elimination occurs when a proton adjacent to a positive charge leaves and generates a double bond.
Because initial formation of a cation is necessary for E1 reactions to occur, E1 reactions are often observed as side reactions to S1 mechanisms.
E1 eliminations proceed with the Elimination of a leaving group leading to the E designation.  Because this mechanism proceeds with the initial dissociation of a single starting material forming a carbocation, this process is considered a uni-molecular reaction.  The involvement of only 1 species in the initial phase of the reaction enhances the mechanistic designation to E1.