This is a big subject. Generally, it's a process where the mash liquid is drawn off below whatever filter mechanism you have (false bottom, brew bag, etc.) and returned to the top of the mash. Generally, it's coupled with a heating method to ensure a consistent and stable mash temp.
There are a number of systems to do this (acronym expansion may be incorrect):
HERMS (Heat Exchange Recirculating Mash System) - With this system, you draw mash liquid off the bottom of the mash tun (MLT), run it through a coil that's immersed in hot water (the Hot Liquor Tank - HLT) and back to the mash. The HLT is heated to a temp that will maintain the desired mash temp in the MLT. A variant of this uses a counterflow chiller device as the heat exchange system by running hot water through the "chiller" making it a heater.
RIMS (Recirculating Infusion Mash System) - With this system, the liquid is sent through a RIMS tube that has a heating element so that the mash is directly heated before returning to the mash.
Kettle RIMS - similar to RIMS except the mash is sent to a second kettle and heated directly before returning to the mash tun
Recirculating BIAB - This is a BIAB system that has a heating element in the mash tun. The liquid is drawn off the bottom and returned at the top to ensure that there is no stratification (layering) of the heat where it's much hotter by the heating element and cooler elsewhere.
The primary advantages if recirculating are:
Stable and consistent temps
Clear wort (it's "filtered" by the grain bed continuously)
Easier step mash (you don't have to add boiling water and dilute the mash to change temps)
Efficiency - with the wort recirculated, you're sort of rinsing the grains continuously.
This is just the tip of the iceberg, but it explains in simple terms what the systems are, how they work and why.