LSP is a path thru the network, PWE is service on top of it, from a CE prospective (devices sending and receiving L2 traffic) it looks like a p2p Ethernet wire, hence the name. In MPLS case (other underlays could be used) LSP provides a labeled path to the remote PE (remote labeled next-hop) that it used to send traffic to (transport label identifies it). In order to distribute/communicate service artifacts control plane is needed. There're 2 different control planes available to do so, LDP(targeted LDP) (rfc8077) and BGP (rfc4761 defines BGP-VPLS vpls (multipoint) but could be used for pw setup) There's also a hybrid solution, where BGP is used for Autodiscovery and LDP for pw setup and maintenance (rfc6074)
These days we are all moving to EVPN. EVPN also supports p2p services (rfc8214) and removes the concept of PW as such.
Hope this helps, Jeff