I'm no expert on machine frame quilting, but I have been doing a lot of research on it over the past few months, because I purchased a system myself just recently. So I'll share what I know (and if anything is wrong, I'd appreciate correction from someone else :) )
Yes, the 15", etc., is the throat space (from needle to the end of the open space; effectively the length of the space to sew in). Your normal sewing machine is usually referred to as a "short arm" (when talking about using it on a frame), and most machines have a 7" throat, I believe -- even the 9" machines like the Juki TL98Q and the Janome 1600P are still called short arms (these are relatively inexpensive machines that can be used on a machine quilting frame). Anything above that, but less than 17" is usually referred to as a "mid arm" (although some manufacturers will still refer to them as long arms). Anything 17" and above is going to be a long arm, and is probably going to cost thousands and thousands of dollars.
The Bailey machines are a nice compromise -- they're big enough that you can do fairly large patterns, but still inexpensive enough that they're not completely out of one's price range. On my 15" (and with my specific frame set up), I can get about 12" of sewing range when I first start quilting...this decreases, of course, as the quilt is rolled up within the throat space. I still haven't done a full quilt yet; I'm still practicing, as I've only had the machine for a few weeks. So far I love it, though -- I'm much happier with it than I was with the Juki that I originally bought...but I wanted to do specific types of patterns which wouldn't be possible in the 9" throat of the Juki.
If you're looking to do frame machine quilting, and you want to do it on a budget, then you don't have a ton of options, from what I can tell. Some people use their regular sewing machine on a frame, which I can't fathom -- you'd have almost no space to stitch as you got towards the end of the quilt. You can upgrade to a Juki or Janome 9" -- but I personally still found that far too limiting, space wise, but I know there are a lot of people that do make it work. Or you can look at a Bailey. There may be other options out there that one could consider "affordable", but this is what I've discovered so far doing my own research.
Some things to think about:
- what's your price range?
- how much throat space do you want? Realistically, what kind of patterns do you want to quilt? If you just want to do small free motion patterns and stippling, then something like a Juki or Janome would probably suit you just fine. If you want to do pantographs, then a larger machine is probably your best bet.
- how much space do you have for a frame? There are a lot of frame makers out there -- Grace, Handiquilter, and Hinterberg to name just a few.
- do you think you'd need a stitch regulator? This is an add-on to the machine that keeps your stitching a uniform length; if you don't have this, then you have to make sure that you always move the machine at a perfectly uniform speed, which can be difficult for a beginner. This is an added expense (usually between $500 and $700, depending on the machine in question)
If you think the Juki/Janome idea would work for you, there are all kinds of packages at allbrands.com, and sewvacdirect.com that include both a frame and a machine. You could get a complete setup with machine, frame, and stitch regulator for around $2000.
Feel free to ask any more questions -- I know I was confused a lot by terminology when I first started researching.