In the most common trailer braking system, your brake controller sends an electrical current to electric magnets in the wheel hubs of the trailer. This magnet then rubs against a plate creating friction which pushes the brake pads against the brake shoe applying the brakes.
In electric over hydraulic brakes, your trailer brake controller sends its electrical current to a hydraulic motor in the trailer which sends pressurized hydraulic fluid to the actuator piston which clamps the brake pads down on both sides of a plate. This is the same type of braking system you have on the front wheels of your car or tow vehicle.
While more expensive, electric over hydraulic brakes can apply a much greater braking force to your trailer than basic electric brakes.
Here is an expellant page containing a full description with photos of one trailer owner's experience in converting their rig from basic electric brakes to electric over hydraulic brakes.
Electric over Hydraulic brakes (external link)