EPGP stands for Effort Points Gear Points. You gain effort points (EP) by show up to ready to raid, showing up on time, participating the raid, participating in boss kills and progression kills, and sitting on stand-by ready to hop in if a spot opens up. When you're on stand-by you recieve 50% of the EP gained by people who are raiding. You don't ever actually spend these points.
You receive gear points (GP) when you win gear. If you look at items that drop in instances/dungeons (while you have your addons turned on) you'll see that they have a gear point value. Some items (usually weapons) will have 2 values. Depending on the class/spec of the person who wins the item, they'll be charged either the higher or lower value of the item. EXAMPLE: A rogue wins a bow. He'll be charged the lesser value of the item because that's not his main weapon, he mostly has it for the stats. A hunter wins the same bow. He'll be charged the higher value because that is his main weapon.
Now here's where people get confused. You don't bid on anything. Your points aren't spent. Instead, what you do is divide your EP by your GP. The number you come out with is called your Priority Rating (PR). The higher this is, this better chance you have at getting loot. New raiders start out with an EP of 0 and a GP of 400. You start with 400 GP as a new raider so you actually have to do some raiding before you're PR outweighs people who have been raiding a lot longer than you

EP / GP = PR
Basic math. And what's nice is the addons track everything for you. You just have to decide what you want to save for and roll accordingly.
The addons you need to get are
EPGP Lootmaster and
EPGP (dkp reloaded). Click their names to be brought to the Curse.com page to download them. All you need to do after that is have them turned on for raids and keep them updated.
BIG EXAMPLE: (Courtesy of Sponty, who is awesomesauce)
Mylindara has 4000 EP and 400 GP, so his PR is 4000/400 = 10.
Gneisha has 4500 EP and 600 GP, so her PR is 4500/600 = 7.5
If a Resto Shaman item drops and both Mylindara and Gneisha roll Mainspec on the item, Mylindara will win because his PR, 10, is higher than Gneisha's PR, 7.5. However, because Mylindara wins the roll, he gets the GP value of the item, which, for sake of example, is 250, making his new GP 650.
Mylindara has 4000 EP and 650 GP, so his PR is 4000/650 = 6.16.
Gneisha has 4500 EP and 600 GP, so her PR is 4500/600 = 7.5
On the next boss, another Resto Shaman item drops, and once again, both Mylindara and Gneisha roll mainspec on the item. Now, because Mylindara's PR has gone down, Gneisha has priority (since her PR is higher than Mylindara's now) and wins the roll (in addition to receiving the appropriate GP).
FAQ:Q: If I'm the only one rolling on a mainspec item, why do I still have to pay for it?
A: Say there's only one holy paladin in the guild. All the spell plate is going to go his way, so he just accrues a bunch of EP and no GP, so his PR is quite high. Now a tier token drops. The holy pally will win it over other pallys, priests or warlocks because they've had to pay for their mainspec gear, making their GP higher and their PR lower. Now that's not fair, is it?
Q: Why do I have to pay for my offspec stuff?
A: Same reason as above, really. Also, say that holy pally wants to roll mainspec on dps gear. Now he's getting his holy gear as "offspec" for free. Besides, offspec gear is given to you for 20% of the original GP cost, which is a really nice discount when you think about it.
Q: I'm new to the guild, and I don't have any EP! I'm never going to catch up to people who have been raiding for months. I'm never gonna get any gear!
A: Well, not for the first few raids you're not. But if you keep showing up you will start to gain a lot of EP, and your GP will remain at 400 because you're not winning any gear. Think about it. You've raiding for a couple weeks and your EP sits pretty at around 6000. 6000 / 400 = 15. Now the other person of your class/spec has 10250 EP. That's a lot! But he's also been winning gear, so his GP sits at 852. 10250 / 852 = 12.04. If you both roll on a piece of gear, you'll win because your PR is higher. We also decay EPGP values every Tuesday, so no one can collect a bunch of EP and just have such a high PR no one can win anything. We also keep an eye on rolls and how people are rolling to make sure no one abuses the system.
Q: How can I check my numbers?
A: You can check your ep/gp/pr numbers by typing /epgp which will bring up a chart. You can also look in your officer note, which would be set up like EP,GP. You'll have to do your own dividing to get your PR

NOTE: If you're not at least a member in rank you won't be able to see these numbers. Ask an officer to promote you.
Have more questions? Post them below and I'll answer them
