Assuming you aren't using NIC vendor proprietary technologies, NIC teaming usually means LACP (802.3ad), which you would also have to configure on the switch by creating a port-channel interface and bundling the physical links to that port-channel by using the channel-group interface configuration command. A common use case for teaming is increased throughput and physical link redundancy between a server and a switch (or a switch stack).
MPIO refers pretty much exclusively to path redundancy between an iSCSI/FCoE/SAS initiator and target. MPIO does not use LACP, and only needs to be configured on both endpoints of the iSCSI/FCoE/SAS connection. A common use case for MPIO is path redundancy between a server node and an iSCSI storage node, where either node has multiple physical interfaces.
MPIO is compatible with LACP if you want increased throughput and physical redundancy for your server-switch connection in addition to path redundancy for your storage connection. For example, if you have four NICs on your server, you can configure two NIC teams (LACP) between the server and the switch, and use MPIO between the server and an MPIO endpoint that will use these two logical paths.