Helen,
There's a whole range of ways magazines finance themselves now. Some do it by donations; some by advertising; some by both; others charge a nominal fee to download, some do that and advertise, etc. Curiously, the higher payers, like Clarkesworld, are often the free-to-read/donations type; so there must be a market there. Having said that, these models appear to work more in SF/Fantasy/Horror than anywhere else, which may be a reflection of the general supportiveness you get amongst fans and writers/artists in those genres. (I donate to quite a few SF mags/podcasts. I've never added it up but suspect that if I factored in overheads, etc, I probably pay more to online mags comparatively than to print ones.)
As for what they pay . . . there is no standard rate; it varies considerably. Many mags don't pay, or pay with copies, etc. The SFWA lists what it considers to be professional magazines; in general, ones that pay at least 5c per word. This is quite a drop on the past, i.e. it was about the same rate back in the 'Golden Age' of SF, in the 30s/40s. So, you wouldn't try to make a living exclusively writing short fiction now; but there are the other benefits I mentioned above. Besides, a 5000 word story at 5c per word is still $250 which is not bad - especially if you can write one in a day or so and don't have to pay an agent's fee; also, it's a direct fee so there's no possible come back from the publisher which you might have if your novel hasn't earned out its advance.
However, it's not entirely down to how much they pay. For instance, the 'big 4' SF/Fantasy print magazines don't pay as highly as a few online mags now do, but in general carry a lot of writing clout and probably still attract the most submissions. Having said that, mags like Beneath Ceaseless Skies and Clarkesworld are rapidly gaining very good reputations for quality.
After the pro level mags, there are quite a few that pay between 1-3c per word; and after that a lot that pay a token fee. However, it's not all about fees: some low-paying mags have very good reputations (e.g. Lady Churchill's Rosebud Wristlet).
You can check out fees, genres, etc at duotrope:
http://www.duotrope.com/
Terry
<Added>Interestingly, just after I posted that, I got an email alert from the forum for the Odyssey Workshop I attended in 2006. It's from a writer who's just made her first short story sale. The story was one she'd put in to that workshop, and has re-written since. The mag pays $20 per story. Which at one level isn't much, especially after nearly 3 years. However, she spends most of her time on novels. She hasn't had any published yet but, although she didn't say it, I suspect selling a short story will probably help her novel-writing confidence no end. There is often a massive gap between starting as a novelist and selling that first novel - a gap that's getting harder to fill at all these days. At least with short stories there is more of a gradated scale, e.g. low-paying to higher-paying, and even the low-paying mags publish stories that can win SF awards.