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BGP (border gateway protocol) is a routing protocol. Personally I consider it sort of a super RIP.
MPLS (multiprotocol label switching) is tag switching, very much like L2 switching. However, it uses MPLS tags, appended to L3 packets (much like L2 VLAN tags are appended to a frame) to make a switching forwarding decision much like a L2 switch makes a switching forwarding decision using L2 MACs; also MPLS tags can be dynamically updated across MPLS switches. As Peter notes, BGP might be used to update MPLS tags.
BTW, a principle initial reason for MPLS was to provide a faster method of making L3 forwarding decisions. It's much, much simpler (and used to be much faster) to deal with fixed size labels than to deal with variable length IP prefixes. Newer hardware now allows traditional routing forwarding to be about as fast as fixed size switching. But, MPLS has other properties, for example, much like VLAN tags can be used to create virtual LANs, MPLS labels can be used to create virtual L3 domains.
