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Mining is the backbone of the EVE Online economy. All player ships, weapons and equipment are built from minerals mined by players. It is not as quick to make money as trading and salvaging can be, at least at first, but it has the potential to be the best and most consistent money-making activity in the game. You can start mining early with ships like the
Caldari Bantam or Osprey, stripping asteroids in high-security space and shipping the minerals where they need to go in a industrial ship like the
Minmatar Mammoth. Though there are many ships that will do the job, these are some of the most effective and most easily available in the beginning.
It is possible to mine by filling up your miner's cargo hold and taking trips back to the station, but this is inefficient until you get a larger ship. As soon as you have access to industrial ships, you should take advantage of a method called jetcan mining - after you have filled your hold on your miner, jettison the ore. This will create a can in space that you can fill extremely full and return to with a larger ship. This is faster and more efficient. (Note: Other players can access these cans, but this will flag them for attack in high security space. This will not stop all of them, so you may want to purchase a portable security container)
After you have your ore at the station, you can either sell it raw or refine it. Whether you do this is up to you, and depends largely on your refining skill. Though it may seem involved, mining is a very lucrative and low-risk activity - a good way for any EVE player to start.
Important skills: Industry, Mining, Caldari Frigate/Cruiser, Minmatar [or your choice] Industrial, Mining Barge, Mining Drone operation, Refining, Astrogeology