LTE networks prefer using the X2 interface for performing inter eNodeB handovers. An S1 handover is a fallback for scenarios where X2 interface is not available.
As the name suggests, S1 handovers take place over the S1-interface. The MME and the SGW are involved during the handover procedure.
An interesting part of LTE S1 handovers is the indirect tunnel that is established to carry the downlink data during the handover process. Refer to the S1 handover call flow for a detailed signaling flow.