L2 VPN vs L3 VPN

With an L2VPN service you connect with your MPLS provider at layer 2. Hence, you do not run any IP services or any routing with your MPLS provider. Hence your two remote sites being connected via this L2VPN service see each other as directly connected at L3 and you run routing protocols between your two sites.

With L3VPN service you connect with your MPLS provider at layer 3. Hence, you will have to run IP services with your provider. Hence you will peer up with your provider using a Routing Protocol and engage in route exchange. The MPLS provider will then send these routes to their remote PE and then advertise these routes to your remote site at L3. At the remote site you would also peer up with L3 to your provider and have this routes advertised to you. So in this scenario, your two remote site routers will not appear as directly connected.

Virtual Private LAN Service (VPLS) is a way to provide Ethernet-based multipoint to multipoint communication over IP or MPLS networks. It allows geographically dispersed sites to share an Ethernet broadcast domain by connecting sites through pseudowires.
A Layer-2 service that emulates a switched Ethernet (V)LAN across a PSN.

Pseudowire

A pseudowire (PW) is an emulation of a native service over a packet switched network (PSN). The native service may be ATM, frame relay, Ethernet, low-rate TDM, or SONET/SDH, while the PSN may be MPLS, IP (either IPv4 or IPv6), or L2TPv3.

VPWS(Virtual Private Wire Service)