Assuming you are talking about MPLS L2 & L3 VPN's the easiest way to think about it is
L2 VPN = think of the MPLS cloud as a L2 switch. Your sites connect into this switch and all sites are using the same IP subnet. So machines at one site communicate with machines at another site using only L2.
L3 VPN = think of the MPLS cloud as a router. Your sites connect to this router and all sites are using different IP subnets. For machines at one site to communicate with machines at another site they have to route via the router.
Attached is a link to the MPLS page which has further links for both L2 and L3 VPNs.
MPLS L3 VPN is like a peer to peer VPN model where ISP takes part in customers routing. They will store the Customers route in their routers and distribute to other customer locations. Advantage is we can have full mesh topology between different locations.
MPLS L2 VPN (also called VPWS - Virtual private wired service) is like Overlay VPN model where ISP will not take part in customers routing. Actually the customer prefixes are transparent to ISP network, wherein SP will just provide the end to end reachability. more of IPLC link. In L2 VPN you cannot have full mesh between customer sites, it will always be a point to point. Advantages for Customer is security because SP will not know about the IP addresses used by the customer and in SP's perspective lot of resources are saved (since SP is not storing the customer routes, run routing protocol etc).
To overcome this, there is a mechanism in MPLS called VPLS (Virtual Private LAN service) where the SP will act as switch and multiple customer sites can connect to the switch. In this case, SP will store the MAC address in their devices.