I have a little compressor that I use for a small brad nailer and blowing up tires. Is there a minimum PSI or horsepower that I need to be able to get to in order to spray paint with it?
Like Eric said, CFM is the important thing, and most small compressors have a low (< 1 CFM @ 45 PSI) output. For doing anything besides light airbrushing (and I mean the artistic kind), you definitely want something with a greater output. In addition, most paint sprayers will tell you what their consumption is, i.e. 1 CFM @ 45 PSI or 2 CFM @ 90 PSI or something like that. It really depends on the airgun because you have HVLP (High Volume, Low Pressure), HVHP, and LVLP (those are rare). But they'll have a sticker or marking of some sort to tell you what they need.
For painting, you really can't use a smaller compressor, you'll need a larger one, probably at least 15 gallons (they tend to have a large enough motor). Just match up the paint gun to the compressor.
As others have said, spraying paint is usually NOT a pressure thing, but VOLUME of air. A typical spray gun will require low pressure, and HIGH volume, FAR more than any standard compressor yields. This one, for example, requires between 0-70 PSI, but 12 CFM. 12 CFM is far more than ANY standard compressor provides, not even in bursts. Or, this one, requires 10-14 CFM. And those numbers are always the minimum. For safety, you need to be higher than that, or the compressor will be constantly working to keep up, or you might need to stop periodically to let it catch up.
Spray guns give you the hint, if they are labeled HVLP. This stands for "High Volume, Low Pressure".
No pancake compressor will give you that volume. No small electric one will suffice. You need something brawny, with a large tank to store the air. A 2-stage compressor will be best here. Don't get something that provides ONLY 12 or 14 CFM. Look for something that will give you at least 16-20 CFM. These compressors are not cheap either.
Next, there is a factor that nobody has stated. I recall that paint sprayers need dry air. Compressed air has water in it. This water will cause a great deal of pain if you don't remove it. So you will need an air filter to catch the water and remove as much of it as you can.
There are also airless paint sprayers. For example, here. I would strongly suggest you look at one of them instead. (I'm not talking about the cheap Wagner models you can buy for $60 at the home centers, but an airless pump that will cost ten times that.)
Finally, there are airbrushes. These are small tools, often designed for fine work. Craftsmen who make models are often the users of these tools, although they can be used in the auto painting industry too. Depending on the brush, small ones can be found that need as little as 20 PSI, to those that need 100 PSI. Airbrushes are typically low volume tools though, so most compressors will suffice. You still want dry air of course, so a filter to separate out the water will be important. (I have a friend who did his air brushing by blowing up a spare car tire at a gas station. Take the spare tire home, and this provided sufficient air for his model work.)
When you go shopping for a compressor with more 'puff' as mentioned above, you should also bear in mind oil. If you run regular tools which are oiled (I oil my nailers pretty much every day), your airlines will be contaminated with oil, probably water too.
You'll need a oil and water trap plus some new lines that you'll only use for painting. A half decent pressure regulator is a good idea too, so you can fine tune the pressure output easily.
The spray gun should have a min/max PSI rating on it (or in the manual), if your compressor can go that high you're good to go. Although if it's a small compressor it's going to be running quite a bit to keep up with the expended air from a paint gun and you may run into trouble if you don't have enough pressure.
Get a compressor a level above what minimum required. You don't want it constantly running and getting hot. I actually tied 2 pancake size together with a T which requires 3 hoses. It worked for spraying lacquer. But very noisy. Lol. Just get a 30 gal with 4-5 hp motor with a moisture filter and you should be fine.