Ok, ok...let me start my saying I have an dry sump system on my race car. And I had to re-engineer it after I bought car. So here goes.
Everything Z06drvr said is 100% correct. However, many classes have crank height rules, usually around 9 inches, sometimes higher, that negates some of the advantage.
Before going further, let me discuss the parts. First, there is a tank. It is usually tall. Oil enters along a edge and swirls inside. It drops through some baffles. In a typical 12 qt tank, the oil in a running engine is just over half way up the take, about 1 ft.
Current dry sump systems are vented through the tank (more on that later) to typically an aluminum tank 3" in diameter and about 6-8" tall. There is a breather vent on top and an oil return on the bottom.
A multistage pump mounted on the engine is driven off a pulley on the front of the crank. Typically, HTD belts are now used. Square-tooth belts used to be used. The pump has one pumping stage. It has a screw adjustment for pressure. Tuning is done to achieve about 5 gallons/minutes on an SBC. Pressure is whatever. Current engines run .020 or so clearances all around, and with oil pumped directly into a main gallery (in contrast to the oil filter mount), pressures are usually around or under 50psi.
The pan has a side kick-out on the right. Size of the kick-out varies. There is also a scrapper closely fitted to the crank (the engine builder grinds this to fit on the completed motor. And a metal screen below and to the right of the crank. The pans can be sheet steel, sheet aluminum, or cast aluminum.
Oil is pumped out of the engine my multiple scavenge stages. Rules in limited classes restrict you to two scavenge stages. Otherwise people run 3-5 scavenge stages. Typically the first two go to the pan, the third (if allowed) goes to the lifter valley, and subsequent ones...well, I haven't dealt with that.
Now to the lines. An AN12 (on an SBC) runs from the tank to the pressure side of the pump. Then from the pump to the oil filter, to the oil cooler (typically in mounted in the hot-side radiator tank), to the main oil galley (on the front above the cam gear on an SBC Bowtie). A tap (fitting with two AN4 lines) may route oil to the valve spring oilers in the valve covers.
The lines from the engine are also AN12. There should always be a screen when the oil exits the engine to catch metal shavings. In either the pump or a fabricated manifold above the pump, the different scavenge lines are merged for the return. Some systems use multiple AN12s for the return. Most current systems use a single AN16. Sometimes another filter is mounted on the return. A couple of tanks come with an Oberg filter on the return.
Back to the pump. I have seen older pumps with 3 1.25" sections. A modern rotor pump may have as small as a .8 section for pressure, and 1.7" sections for scavenge. Pumps are wither a rotor or gerotor design. Both in use, although the pump I think is the cats pajamas is a rotor design.
If you think you think about, you can't return more oil than you pump. However, you can return air. A vacuum in the the crankcase can add serious hp. I believe Nextel cup cars draw close to 20in.
To get vacuum you need to seal everything. Different front and rear crank seals. Distributors (on cars with them). the lifter valley. And pumps are now designed for it. The Barnes pump uses two rotors instead of the more common 8 or 10 (it is too late to go into the garage to count), because it sucks more air.
You buy the parts separately. Moroso, ARE, Milidon, Stefs, Canton, etc. make pans. Barnes, Peterson, Moroso, Johnson, SCP (in CA, not NC), and others make pumps. They are usually sold with the mount. Pulleys are part of complete systems from Jones and others. Peterson and others make the tanks. Numerous people make the breather cans, however I like Oval Craft. All of the radiator (C&R, Griffen, Fluidyne, etc.) people make oil coolers in the radiators.
Peterson (petersonfluidsys.com) also makes one way check valves for the breathers (there is also a firm in Florida whose name escapes me). They come to ways. To vent of the pump fails to scavenge enough air. And to vent air in if scavenges too much. I have the first. Nextel Cup cars have both.
There are two other nice benefits. If you go off road (which happens in racing more than we would like to admit), the shallower oil pan (further off the ground) is less likely to develop a new drain hole. And using neat little bolts from peterson to hold the pulley on the pump, you can easily prime the engine with an air ratchet.
Typical Tank:
Typical pump:
Typical pan...actually an LS1 pan: