The difference between a level designer and environment artist. The level designer only cares about volume while the artist worries about the way it looks.
The easiest way to describe the job of a level designer would be: Designing the path the player has to/ can follow to go from point A to B. How the environment looks like is not his job however the placements of assets, power-ups ( in any form from ammo to health etc )
Enemies, NPC'S, duration of the level are all part of his job. A level designer often get's told what sort of level he will work on, the assets available , the purpose and story behind the level and can then start blue boxing a level. Blue boxing is a term used for making a level out of basic shapes
to get a rough idea of where the players can go and how they will reach certain places it's much like an interactive blue print of a level. Most engines will have some easy way of letting people blue box without having to start 3D software.(p.s people use different terms for this)
Some blue boxing i've done a long time ago.
Another possible job of a level designer could be to design certain obstacles that occur in a level. This could either be part of the Games design or part of a different reason such as slowing players down from progressing too fast through a level.
Valve is one of those companies that do their level design very well. They break levels up in arena's where players fight off monsters and switch them with puzzles to calm players down and give them some time to recover /restock. Another thing Valve has done really well when it comes to level design is in Left for Death 2. The artificial director decided when to spawn zombies to make players move faster ( or slower) what zombies to keep things interesting and challenging but is also able
To make small changes to the level by opening or closing parts of the level. this makes the level re playable and unpredictable. A good example of a game where this has been done poorly would be Diablo 3. The levels pretty much stay the same the variety in monsters is minimum after the first play through making it quite repetitive.
Depending on what sort of level is being designed there's a few things the level designer needs to keep in mind when it comes to a Single player level it's often : How hard is the level compared to the ones before / after.
What skills will the player require to complete the level ( have they been taught or do they need to be introduced), How much time does it take to complete, Is there a story involved and if so where ? is it consistent are there any glitches and if so how can they be fixed, where will the power-ups/weapons/health/items etc be and probably most important is it fun to play.
Multiplayer is equally as hard if not harder ( depending on the game) as there's a lot of balancing questions that need to be asked: Are both sides ( if there are sides) equally matched, What sort of level will this be ( in first person shooters it might be a sniper level or trying to prevent sniping in general, based on
the level design this can go either way) , is it repetitive and/or replay able, where will the power-ups/weapons/health/items etc and of course is it fun ?
It's an important thing to plan a level out carefully and do plenty of play testing to get feedback on what works and what doesn't as this often affects the end result.
Some small examples
Planetside 2 is a Free to play Masive Multiplayer Online First Person Shooter ( mmo fps)
It's an interesting example as it combines several level design decisions together.
As can be seen there's a lot of open space making it perfect for Vehicular combat ( planes/tanks etc)
And for snipers ( this guy did not have a good day )
However there's plenty of cover ( if he would have only used this) making it possible to both possible to
snipe and fight in close/medium range. these are several decisions made by the level designer.