Regenerative Drives are drives that operate in all 4 quadrants of power.
Quadrant I - Drive delivers forward torque, motor rotating forward. ('Motoring Mode' of operation)
This is the 'Normal' condition, providing power to a load, or accelerating. Similar to using a simple motor starter.
Quadrant II - Drive delivers reverse torque, motor rotating forward. ('Generating Mode' of operation)
This is a 'Regenerative' condition, where the drive itself is absorbing power from a load, such as an overhauling load or decelerating.
Quadrant III - Drive delivers reverse torque, motor rotating reverse. ('Motoring Mode' of operation)
Basically the same as Quad I, except we are going backwards. Similar to using reversing starters.
Quadrant IV - Drive delivers forward torque, motor rotating reverse. ('Generating Mode' of operation)
The 'Other Regenerative' condition, where again, the drive is absorbing power from the load in order to bring the motor towards zero speed.
A Regenerative drive accomplishes all of the above purely electronically, without having to use contactors to switch leads around.
DC Motors/Drives are simpler, so I'll go into them. Assume a 3 phase, fully controlled bridge, drive here.
A Single quadrant DC Drive would have one power bridge of six controlled switching devices, (usually SCR's) to control the applied voltage level to the armature of the DC Motor. This type of drive can only run in a motoring mode, and would require physically switching armature or field leads to reverse the torque direction
A Four quadrant DC Drive would have two complete sets of power bridges, or twelve controlled switching devices. One bridge controls Forward Torque, and one controls Reverse Torque. During operation, only one set of bridges is active at a time. For straight motoring in forward, the FWD bridge would be in control of the power to the motor. For straight motoring in reverse, the REV bridge is in control.
What happens though, when we want to decel from forward rapidly, or follow a controlled ramp for example? If the drive just shuts off the FWD bridge, the motor would coast down to zero based on the system inertia. This is an uncontrolled stop. In the Regen (4 Quadrent, 4Q) drive, the controller switches off the FWD bridge, and begins turning on the REV bridge, in order to actively generate a negative torque to the motor, allowing for a controlled rate of decel. Since the motor is now acting as a generator, with the drive as the 'load' power is going somewhere. In the case of the DC Drive, the power is dumped directly back onto the incoming supply lines, and is thus mostly recovered (except for wire/switching heat losses).
AC Drives can be regenerative also, but are generally move expensive, as AC Drives require two full "DC Drive Type" bridges between the incoming AC Source, and the drives internal DC Bus. The internal DC Bus is used to provide power to the six output devices that actually create the rotating field for an AC motor.
Non Regen AC Drives most commonly have a simple 3 phase full wave diode bridge to generate the DC Bus. Regen drives must have instead the equivilant of an entire regenerative DC Drive in place of the simple diode bridge, hence the added cost.
Non-regen AC drives, almost without exception, can always operate directly in quadrants I and III (Motoring forward or Reverse).
AC Drives that use a chopper and resistor to enable Quadrant II and IV operation aren't regenerative in the sense of delivering power back to the line, but are regenerative in that they can actively remove power from the driven load.
Regen drives are most often used where there are many and rapid reversals of torque direction, or for overhauling loads, or say, for unwinders, where you are holding back against the rest of the machine.