During X2 Handover procedure, the target eNB sends the path switch request message to the MME to inform that the UE has changed cell & the SGW needs to forward the incoming packets to a different destination.
So the MME sends a modify bearer request message to the SGW to update the downlink TEIDs & ip address.
The SGW is supposed to use this updated TEID (tunnel end point identifier) & address to forward any incoming packet to the target enodeB.
The SGW changes the downlink data path to the target enB. But before that the SGW sends the end marker packets on the old path to the source eNB informing that these are the last few packets you are receiving after which the incoming packets will be sent to the target enodeB.
SGW responds back with a modify bearer response message to the MME.
The MME responds back with path switch ack message.
Now the target enodeB informs the source enodeB about the successful handover by sending a UE context release msg so that the source enodeB releases all the radio and control plane related resources associated with that particular UE.
In short you can say that the path switch request procedure is used to inform the SGW to switch the path of downlink packets & re-route the incoming packets to a new destination.
Hope this helps!