When I got into the industry way back in the 1980s, there was one type of game designer in our industry. We called him "programmer." He (or she in the case of the rare few like Dona Bailey) was often a one-man show responsible for design, programming, sound, and art all in one. I worked with one of these individuals right on up until 1988, in fact. Eventually, games grew larger, and with the increased size came a specialization of tasks. Teams formed, and we had artists, programmers, designers and even a sound person. Eventually producers came along to network and schedule the increasingly growing teams.
In the last ten years, but more so in the last five, we've seen greater specialization within the fields themselves so that now, and at least in my field, the term "game designer" sounds general. It's a perfectly okay term to use on indie game projects, but when you're talking about a big Xbox 360 production, saying, "I'm the game designer", is likely to result in a follow up, "Yeah, but what exactly did you design?"
We've got roles now. I've listed them below as I've experienced them. Bear in mind that there's no such thing as OSHA requirements for design gigs, so what I experienced in these various roles will likely gust in one direction or another depending on the company. As usual, there are few absolutes in the game industry.
Lead designers are responsible for managing a team of designers and making sure that the game's overall vision is achieved. As a lead, you will put out fires, do what needs to be done, maintain a good attitude and teach those working with you the ropes. By yourself or with the help of others, you will establish the game's core and define its feature set (or it may be defined for you by publisher mandate). You may decide how the project is documented, determine the basic systems and create the overall story arc, or you might assign these to another designer on the team. You review what they did and make sure it integrates into the bigger picture. At some point or another, you may take on one (or two or three) of the roles listed below. You also need to develop an eye that can analyze and critique work, your own along with everyone else's.
You will work a great deal with the art and programming leads to make sure the feature that you're hoping for is feasible technically and artistically. You'll hear "no" a lot. As a note, teams go much smoother when the leads are a tight group. If there is lead warfare, it's painful for all involved. Fortunately for me, I've only experienced this once a long, long time ago. The producer squashed the art lead before it reached a dramatic point, and all was well.
New leads or those hoping to be lead someday may confuse the term "lead" with "dictator" or "visionary." It is neither. A good lead knows when to get out of the way and encourages others to help shape the game. Good games are rarely the "vision" of a single individual, but rather a collective whole. I once heard a quote which I will paraphrase (badly) here: the job of a lead is to encourage people smarter than they are to do their jobs well.
Level designers are probably the best known of all game designers, and it's their job to create the level-by-level play in a game. If you've ever jumped when a monster nailed you coming around a corner, discovered a particularly advantageous place to shoot from, or felt tension coming out of an in-game elevator, know that somewhere, there's a level designer who planned that experience for you. Level designers generally place the creatures, items, props (boxes or crates or whatever) within the level, or they may have junior level designers who do it. Level designers are usually artists and programmers-of-a-sort as well, and may create many of the assets needed to finish the level and do all the necessary scripting to make things go exactly as they desire them to.
Within the level design field, there are different specialties as well. FPS level design is different than RPG or MMO level design, for instance, and each type requires an in-depth knowledge of the standard play mechanics and level flow within those games. I've worked on RPG level design my whole career.