Mining in EVE Online is a relatively simple affair. You fly out to one of the many asteroid belts, target a rock, start your mining laser, and wait. Every minute (or every 3 minutes if you have a specialized mining barge with a strip mining laser) a load of ore is delivered to your cargo hold. Voila, instant wealth! Only of course the amount of ore you can mine in a minute in a totally safe space isn't worth very much.
Once you move away from the sectors with the highest security level, you will find NPC pirates in the asteroid belts. They don't respawn very fast, thus you can go to the asteroid belt with a combat ship, shoot down the pirates ("rats"), go back to the station, switch to your mining ship, and go back to the asteroid belt to mine for a while until the rats are back. But once you enter space with a security level of less than 0.5, your problem will mostly be player pirates, against whom that method obviously won't work.
The other problem of mining is cargo hold. Except for specialized mining barges (If I wanted to fly one, I'd need to spend over 10 days of skill training for the minimum required skills), ships in EVE Online come in two types: Ships with large cargo hold which have zero or one turrets on which to install a mining laser, or ships with several turrets which have a small cargo hold. If you take the former, you'll get very little ore per minute, but in a safe spot you can mine afk for an hour until your hold is full. Not efficient, but on my dual screen setup I can put EVE on the second screen and do something else on the main screen. The more common method is to use a ship with several mining lasers, in which case your cargo hold will be full after a few minutes. Then you have two options: The slower, but safer method is to fly back to the station, unload, fly back out, and restart mining. The more risky method is to "jettison" your ore into space, which will create a cargo container or "can" floating next to you in space which has a huge volume. So you mine, and every couple of minutes you shift your ore from your cargo hold into the can. Once the can is full, you fly back to the station, switch to a ship with a big cargo hold, fly back to the can, and get the ore. But cans aren't safe, anyone can come and steal your ore, if he doesn't mind that this flags him as being attackable by you and your corporation (guild) for 15 minutes.
Ideally you'd have a combat ship, a mining ship, and a ship to transport the ore with. Lots of EVE players have multiple accounts, and run such an operation by multiboxing. But at a larger scale many corporations organize mining events, a "raid" on an asteroid belt so to say. Some players mine, others transport the ore, and a third group is in combat ships keeping everyone safe. Corporate mining is pretty much the only way one can mine in low sec space.
I participated in a corporate mining operation Sunday evening, for four-and-a-half hours. Due to me new to the game I only had a small frigate to mine with, with two mining lasers, and a cargo hold that couldn't even hold three minutes worth of mining (and that with modules to increase cargo hold). So every two minutes I shifted my ore from the cargo hold to the can. When the rock I had targeted was empty, I shifted to the next rock, and when the asteroid belt we mined was empty, we shifted to the next belt. That was all. 4.5 hours of clicking every two minutes to move ore from the hold to the can. Incredibly boring as a gameplay activity. Of course pretty much everyone in the mining operation was bored, so we had lots of time to chat. It reminded me a bit of Everquest, where the developers deliberately forced groups to "meditate" for minutes between combats to regenerate mana, to encourage groups to chat. I understand the idea, but already back in EQ that sort of downtime to enforce social interaction wasn't really popular. And personally while I do like to chat sometimes, over 4 hours of chatting is a bit too much for me. So in case you noticed yesterday had two rather long blog entries: Those were written while mining Sunday night.
The one redeeming feature of mining in EVE is that whatever you do, you produce something useful. Even if you are in your newbie ship on the tutorial mission to mine Veldspar and refine it to Tritanium, the Tritanium is not just used for "low level" items, like mining copper in World of Warcraft would be. While not extremely valuable, Tritanium is used for manufacturing pretty much everything, so there is a solid market for it. The further you advance in mining, to rarer ores, the more advanced metals you get, but Tritanium is usually still a part of the mix of metals refining advanced ores gives you. Thus unlike WoW you don't have a situation where low level resource gathering only produces material for low level crafting of items for low level characters. Much better economic integration of new players is a good thing. But mining in EVE is still boring.