L2 switches deals with L2 frames (e.g. Ethernet frame with MAC addresses).
L3 switches deal with L3 packets (e.g. IP packets and IP addresses).
L2 switches usually learn the frame's outgoing port by seeing earlier frames on the same port with the destination address. If destination address has not yet been seen, L2 switches usually flood the frame to all other ports except the one it arrived on.
L3 switches usually learn the packet's outgoing port by manual configuration (e.g. static route), from a dynamic routing protocol (e.g. RIP/OSPF/EIGRP/BGP), or by known connected destination address blocks (e.g. interface addresses with subnet masks). If the destination address is unknown, normally the packet will be discarded unless a default path has been defined.
A L2 switch is also known as a multiport bridge.
A L3 switch is also known as a multilayer switch. (It often can function as a L2 switch too.)
A L2 swich is commonly used in place of a hub.
A L3 switch is commonly used in place of a router.
The difference between a L3 switch and a router is the former is often "faster" but the latter is often feature "richer". So, L3 switches are common within LANs and routers common for within WANs. (Today the two often overlap.)
