EVE Online is a MMORPG in space. You are represented by your space ship and a pilot portrait.
Characters are based on individual skills, and skill training is time based. You set a skill to be trained and in a certain amount of seconds, minutes, hours, days, whatever, then the skill will level up. You may only have one skill training at any given time. Skill training still continues when you are logged off. Therefore, there is no need to grind monsters for experience to reach some level cap. Effectively, there is no cap to a character because it would take many many years to train them all. It is, however, also relatively easy for newer players to catch up to the veterans because each ship can only use so many of your total skills at once (maxed out skills in Amarrian Battleships won't help you much when you're in a Gallente Frigate, after all), and there is a cap on how high a skill can be trained.
Further, the time to train a skill increases for each level -- training level 1 Frigates takes half an hour, while training from level 4 to level 5 might take a week -- while the benefits remain the same. So a new player can train a skill through level 4 relatively quickly and recieve 80% of the benefits that a maxed out veteran is recieving from that same skill. This means you that new players can focus on very particular skills and become productive players nearly from the start, while veterans have often mastered a handful of areas and are proficient in several more.
EVE is a sandbox with the opportunity for the player to do anything they would like. There are no professions that are set in stone, so if you'd like to mine one day and then kill some people who are mining the next day, then so be it. Professions are labels that are acknowledged by the player and not the game. Many people ask how they can become a pirate or bounty hunter, but in reality of the game, the player is whatever the player believes they are.
There are few restrictions on PvP which means non-consensual combat is encouraged, although there are ways to be safe, as well. If your spaceship is destroyed, then it is gone. You would have to buy another spaceship; however this also applies to other players that you kill (it is possible to buy insurance to recover some cost if your ship is destroyed). There are regions of space like high security Empire space that are relatively safe -- peacekeeping officers will pursue and destroy any player firing on another ship -- and regions of space like 0.0 security space that are no-holds-barred battlefields. However, nowhere is ever COMPLETELY safe.
So with spaceships blowing up left and right, EVE has a player driven economy and market. It can be profitable to acquire minerals and make ships or ship modules (equipment) using blueprints, and sell to the market. Or you can just mine for minerals and sell those to players who manufacture that are too lazy to mine for themselves. There are many different ways to get money.
The world of EVE is on a single server, which means every subscriber is playing with every other subscriber. Currently, Eve often has between 15,000 - 30,000 players playing simultaneously. If you're (in)famous in the server, you're (in)famous throughout the entire game.