Sparging is the process by which all the residual sugars remaining in the mash are washed out of the grains and into the boil kettle. There are two main methods of sparging, batch sparging and fly sparging.
Because I batch sparge, I end up having to lauter again during each sparge. Once I have the mash tun empty after lautering, I close my spigot and add hot water. I double batch sparge, so I do two separate sparges. Normally I'll add about 8 quarts of 180F water for my first batch, and 8-10 quarts of 180F water for my second batch. This help to get the mash up towards 167F, known as "mash-out" which halts enzymatic activity, lowers viscosity of the wort, and assists in washing out as much sugar as possible. For each batch, I dump in the water, stir up the grain bed, then begin the lautering process again before proceeding to allow the rest of the wort into the boil kettle. Once the sparging is complete I can begin to boil the wort.
Although I use a bazooka screen, that isn't the only way to filter out grains. Some homebrewers will use plastic coolers as mash tuns due to their insulating capabilities, and will rig together grids of copper tubing with little holes in them to act kind of as a large screen. Another common method is to use a false bottom above the drain spout. Grain collects on top of the false bottom and the slots cut into the false bottom allow wort through without allowing grain particles to pass.
Fly sparging is a continuous sparge. Once lautering is complete and the grain bed has begun to filter, hot water is added from above at a rate equal to the outflow of wort from the mash tun. This can be done in a number of ways, but generally requires some sort of setup to spread the sparge water equally across the surface of the grain bed. As more and more sugars are sparged out of the grain bed, the pH of the wort runoff rises and the specific gravity falls. At some point (I've read at an SG of 1.010 or 1.019) the low specific gravity and high pH cause tannins to be leached out of the barley hulls, leading to astringency in the wort and thus the beer. So the stopping point for sparging is whenever the gravity of the wort begins to get too low.
Advantages and Disadvantages of Each Method
The advantage of fly sparging is that it is more efficient than batch sparging. Because water is being spread equally on the surface of the grain bed, all the sugars throughout the grain bed are being washed out, and there is no channelling taking place. With batch sparging there is often channelling of some sort. In my mash tun I notice much of the wort draining around the edges of the tun, not necessarily evenly throughout the grain bed. That being said, for homebrewers the efficiency gains from fly sparging are probably only on the order of a couple percent. While small gains can be commercially significant, for a homebrewer they're probably not worth the effort.
The advantage of batch sparging is that it's easier than fly sparging, and it allows you to use a precise amount of water. Rather than trickling water through until the runoff reaches a certain gravity, batch sparging allows me to calculate my water needs in advance. I know that my grains soak up about 0.130 gallons of water per pound, and I know how much wort I lose by boiling, so know that I need about 8.25 gallons of wort into my boil kettle. Mash Water + Sparge Water - (pounds of grain * 0.130) = 8.25 gallons. So if I have 10 pounds of grain and use 5 gallons of water in the mash, I know that I need to use about 4.5 gallons of water in sparging. I suppose I could heat 4.5 gallons of water and slowly fly sparge it, but for me it's easier and far less time consuming to batch sparge. I'd rather lose a couple points of efficiency than an hour out of my day.