I mean, what qualifications do they look for? Do you already need to be a published columnist or something?
Well, it varies greatly. Some think-tanks run blogs as well, and will hire sufficiently talented people straight out of college (or nearly so) to write for them. And if you’re a student, of course, there are internships - highly competitive ones, in most cases.
For more senior positions, of course, they’ll often look for people with more advanced degrees, or experience in the field. Brookings, Heritage and AEI, for example, all have reputations for being the places that people of the appropriate party hang out when they’re in the loyal opposition.
The only real rule of thumb for the top-tier thinktanks is that, before any other qualification, you really do need to be something of a badass in your field. Short of that, you really needn’t bother - DC is full of amazingly talented, smart, ambitious people. That’s your competition, along with all the smart, ambitious folks elsewhere looking for a DC gig.
Totally talking out of my ass here, but I’d imagine that most employees of such places are culled directly from college, lobbying groups, and other consulting companies. In other words, you get recruited.
There are everything from entry-level to senior positions that one could apply for. Entry-level folks can be right out of college and just have an interest (and maybe a major) in a field relevant to what the think tank works on. The more senior you are, the more experience you should have in those fields.
That experience is typically academic or governmental work in the field, though there are occasional journalists. There isn’t any particular job requirement, for example, someone who worked in, say, the Department of Education for a number of years is generally as qualified as someone who just got a PhD in education policy.
Think tanks are also good “bullpens” for Washington, DC insiders whose political party happens to be out of power at the moment. So, let’s say you’re a Republican working for a member of Congress who lost his re-election in 2008. A staffer might join a conservative think-tank for a couple years until more Republicans get elected to office.
ETA: and applying for a think tank job is like applying for any other job. You could get a job by responding to vacancy announcements or if someone at the think tank happens to know you professionally.