As stated, it depends on the company you're working for as to who does what. Some companies don't even have "game designers", just a programmer designated to work out the gameplay assets. Personally I tend to disagree with this way of thinking, but that could be because it would be my job on the line. :P
The titles that you're talking about such as character design are generally done by general designers working under a lead designer or a creative director. The larger the team, the more of a chance you're going to get people doing specific things, and of course for a smaller team, not all of these positions exist, but the structure I have seen is:
A lot of teams treat this rather informally, though. For example, on many projects, everyone on the design team will brainstorm and collaborate on certain ideas for gameplay. During the concept phase, you need as many heads as possible to make something work out, and the design team will hold meetings in which they do nothing but toss ideas off of each other. But in the end, the Creative Director or Lead Designer has the final say. But again, that's not to say someone like a Junior Designer isn't going to get to do fun stuff, but they're not going to make the executive decisions.
Of course being a designer isn't all about just sitting in a circle and thinking up ideas. There's a lot of documentation, the largest of which is the GDD, a comprehensive "bible" for the entire development team. Designers are also responsible for the concept document as needed to pitch to publishers, etc, character worksheets, prop writeups, level documents, etc. In some teams, all of those specific sheets are appended to the GDD, but in others they're left out and just passed around to whoever needs them. Again, a lot of procedure varies.
I want to say that when you see a call for a Level Designer, though, it's not often looking for the type of designer you're thinking of. Most calls for a level designer are looking for someone who can use the provided engine (Unreal, Torque, Gamebryo, etc) and create the level within it using prefabs and other fun stuff.