Level design could be seen as an off-set to environment art. It's the creation of the level; when the designer pushes the player towards the level's goal. Obviously every game gives the player a goal to achieve.
Firstly a designer might start off with mapping out the area you explore in (the level). What the terrain is like, where the river goes and if that hill is in the way or not. That sort of thing. To help kick start this, some may find it easier to create a simple geometric blocked layout to see how everything might look. This section would come under environment design.
In my opinion, the main body of game design is the goal; the purpose of why you are playing the game, that thing you're trying to get to. Most games have a story line and set achievements throughout. This gives the player some sense of accomplishment, and keeps them interested throughout their progression. Not all games have a set storyline, though. As games have developed, optional story lines have become more and more common. Back when I got my first PlayStation, I couldn't take Crash Bandicoot out of the jungle run and play on mini games. Now, on games such as World of Warcraft, Skyrim and the GTA series, players are allowed a free reign and can take their characters in what ever direction they like. A Warrior on Wow could be a herbalist, alchemist and first aider for example.
The set of the goal fits in with the story, it should be unique, original and interesting. It has to drive the player to want to play the game and to continue, otherwise you have an unsuccessful game that just won't sell. (it's all about the money...)
The goal has been set but it should be made specifically to pace the player through various activities in which the player is pushed and progressed towards the goal. (initially this is the game)
Below, are games to give a basic idea of what the heck I'm talking about. For example; Gex the Geko for Playstation 1 was a fairly simple game. The sort of, kill the badies get the remote controls to unlock new television levels, then kill the big baddy that stole your hot model girlfriend. So you complete all the mini activities which are the levels to get to the end of the goal which is to rescue your girlfriend. Simple.
But in these new open world exploring games such as; world of warcraft, diable 3 and skyrim. It is all the same for them. There is still a set start and finishing point towards the goal. Except it is different in the fact that you aren't pushed to do everything in a certain way. You can ignore the main quests and run around doing your own crazy thing but in the end you will have the option to complete the goal/endgame. (yes i'm finding it really hard to explain what I mean)