In simple terms, single action means that pressing the trigger works the air and the paint together, with no further control apart from regulating airflow at the compressor only. The flow of paint has to be pre-set at the nozzle, to the desired spray pattern area.
Dual action has much more precise control - pressing the trigger works the air, and pulling back the trigger controls the paint flow. The further back the trigger is pulled, means more paint will flow, and therefore a wider pattern.
The single action brushes are really more of a mini spray gun, and although perhaps a little limited, can still meet many needs, depending on the size of the nozzle/jet. They are particularly good for covering larger areas, or overall single colour or clear coat work, and are normally syphon, or suction feed, from a jar below the actual brush.
Dual action are more precision instruments, true airbrushes, and in general, are capable of finer detail work which, depending on the nozzle type, and distance from the work being done, can be a pencil-thin line if required. In most cases, they are gravity fed, meaning there's a cup or reservoir mounted on top, or on the side, of the brush body.
If you have a look at web-sites specialising in airbrushes, you should find that the specification details will show things such as the range of operating pressure, and the range of spray pattern. For example, the latter might be something like 'From 0.5mm to 15mm'.
As for compressors, whilst it's true that a compressor with a tank is
desirable, for the reasons stated in the previous posts, it's not absolutely
essential in most modelling jobs. The reason for this is, that, generally, the actual time of each 'burst' when spraying a model (depending on size of model of course - might differ with very large scale stuff) is relatively short, and in most cases well within the capacity of piston or reciprocating compressors.
That said, as with any other tool, it's generally a good idea to buy the best you can afford (given the specs are suitable), which doesn't have to be the most expensive.
Traditionally, most airbrushes and compressors were designed and built for art, or graphic arts studio work, where they'd be banging away virtually all day, every day, 5 or 6 days a week, which of course meant that high standards of quality and engineering were required, and a higher price tag because of this.
However, in modelling, the demands on the equipment is nowhere near as harsh, and the average set up is way over what is actually needed.
Nowadays, due to the uses of airbrushes and compressors in such diverse areas as modelling to beauty salons, the range of choice in brushes and compressors is much larger, and prices much lower, than they were even 10 or 15 years ago, so it's worth spending some time looking around, learning a bit about the specs from published info at various sales outlets, and comparing prices, especially package deals.
I don't have any info handy on sales outlets in the USA, but a quick look at the UK based site below might be a good intro or looking at specs and so on.
Have a look at
The Spraygun Company Ltd
This is the place I got my current 'brushes from (their own AB brand, which are excellent, especially for the price).
Hope this helps.