X3 is like a spare empire sim, more or less. You start as a new pilot in a dinky ship, and then you can do pretty much whatever you want. You can start a galaxy-wide trading empire, you can do missions, you can manufacture your own goods (and later, ships), you can even pirate. There are bigger and better ship types to fly, and several alien races to pick from, each with their own ship lineup with different stats.
In X3: Terran Conflict, there are also special non-purchaseable ships around the galaxy that you can capture. Obviously, the people currently occupying them won't be happy. The way capturing works is you have to bring the shields down, then launch your boarding party, and keep the shields low while they clear various sections of the ships. The speed, chance of success, and resulting hull damage from the capture depends on the skill level of your away team, and if the shields regenerate above a certain (low) amount while your crew are still cutting through the hull, they'll end up going all crispy-like.
There is a main "storyline" in the game and other mission arcs (such as the one that gives you the ability to build your own ships). Otherwise, it's a pure sandbox. You can do whatever you want to earn money, and spend it however you want. You can buy ships and create your own fleet (AI takes over your other ships and you can give them orders through command menus). You can buy a fleet of trading ships and set up automatic routes, or hire pilots to trade for you, they seek out better and better opportunities as the pilots level up. You can build your own stations to manufacture and sell, and pretty much everything can be automated.
You can also, eventually, buy/capture capital ships, and unlike most games, in X3 a Carrier capital is actually a carrier, they can house LOTS of normal-sized fighters (like the ones you start out flying). Of course, you have to own those, but that makes the carriers pretty scary.
With all the freedom come downsides, however:
1) The only tutorial you get is how to physically fly your ship. That's it. To figure out how to build stations, fit your ships, etc, you basically have to go look online.
2) There's nobody to play against. There's no "AI". The big empires are just completely automated and nothing really ever happens. They'll happily let pirates invade their system and blow up stations and not even try to save them unless some of their ships are close, at which case they'll just automatically engage and (most likely) die. This basically makes space feel empty except for you, because nothing really goes on that you don't get involved in.
3) The UI is.. not the best. Granted, it's a complex game, and the UI is much improved in Terran Conflict over the original X3, but it's still a bit messy. It will take you some time to figure out where everything is.
4) You should definitely be prepared to spend a good deal of time reading all sorts of guides and tutorials online.
I'll echo the comment above, if you're going to get one, go straight to X3: TC. It has basically everything X3 has plus better UI/mission system/more ships and a new start (Terran).
