WTF, Gaz? We don't want any of these people working in our industry...don't give them any help. I know you're all snug and secure on your prison island, but, please, think of your friends!
ANYWAY...
First up, people are being too general with their terms here. There are multiple types of designers, and unless you've got a really good connection or are headed to a really small company, you're not going to be a "game designer" right away. A typical design heirarchy goes something like this...
Level Designer
Game Designer
Senior Designer
Lead Designer
Game Director
Creative Director
Level Designers make levels. Game and Senior Designers oversee the development of the game's ideas and do some direct implementation themselves, often times focusing more on tuning of systems rather than individual levels. Lead Designers oversee the entire design department, and usually have some managerial or scheduling tasks folded in with the design stuff. A game director, should the project have it, is the holder of the project's vision and works with the design, art and programming departments to realize that. A Creative Director oversees the development of several projects.
I'm a game designer. I got in through connections I made while I was gaming media, which lead to two summer design internships and a job after I graduated from college. This is not a traditional path to design, of course...
A more traditional route would be through QA testing. Many companies hire designers from their QA departments in part because design is generally underappreciated, and QA people are generally hard-working, are forced to keep a schedule and have to communicate effectively with a team in order to do their job. I guess this is what they call "paying your dues," and it doesn't always leads to good designer candidates, but being a tester and proving your worth there while you play around with mods and design tools is generally a good way to get a job as a Level Designer.
While an understanding of games is helpful, of course, the fact of the matter is that for most people it's not going to matter because in most situations you won't be making any real, overarching gameplay decisions or contributions for a while - understand this now or be unhappy later.
To move beyond Level Designer, the most important thing is an ability to communicate, both verbally and in writing. You have no idea how infuriating it can be to work with someone who can't communicate a simple idea about a game, and how much chaos that can create on a project. I've worked with leads who couldn't put a coherent sentence together to save their lives, and seen his design team's intent get twisted and raped because of it, seriously hurting the quality of the game. Some of this is bound to happen just through standard project compromises to meet the schedule and whatnot, but when it happens simply because your lead can't talk good it's really painful.
With all of that out of the way, I should say that I'm currently pretty miserable at work, and have considered getting out of design and the games industry several times. This mostly stems from a bad string of experiences than the professional as a whole, but companies do tend to put technology and process before game quality. That said, design is not afforded the respect it probably deserves because of the simple fact that we can't really make anything without other people's help. Additionally, everyone thinks they're a designer and are constantly bombarding you with bad, undeveloped ideas, and think that because they can do it your job is meaningless. Of course, at the same time, they'll tell you that they can't do their work without whatever it is you wrote. And even though programming problems might ruin a game, ultimately the lion's share of the responsibility falls in to the designers' lap when it comes time to look for a new job.
Hope some or all of this is useful.