I must own the cheapest cutter here, the ubiquitous K40 (£340 new, from Ebay). I find it ideal for model work, if I was running a business then one of the larger and much more expensive lasers would be my first call. But the K40 handles anything I can throw at it for modelling purposes.
It cuts styrene,
It cuts ply,
It cuts solid timber.
It cuts acrylic, both cast and sheet.
It will engrave on all the above and also glass, marble, slate, brick any material really - bearing in mind some man made materials outgass some very toxic substances, so extractor units are a must.
Plywood I once cut a piece of 10mm thick just to see if it could - it did with multiple passes. Think of a laser like a saw blade. You push the saw and it removes a layer of timber as sawdust, pull the saw back and repeat, another layer of sawdust and the cut gets deeper.
The laser works in exactly the same way when doing multiple passes, except instead of producing sawdust, the material removed is vapourised and dissapears up the extractor as smoke and fumes.
Also unlike handtools as gunner mentions, what you draw on the computer, you get from the laser - it just doesn't make mistakes, no overcuts, loose joints or anything. As long as you can use a drawing program, Coreldraw for intance, you can get results from a laser cutter. The smaller laser like mine takes all sorts of files for import, dxf, cdr even bitmaps. Take a look at the tramp steamer in my byline, almost every part of this ship was done on the laser, the ships wheel which was only 5mm in diameter complete with spokes was cut, something I could never accomplish by hand alone.
The only drawback for me is that it doesn't cut metal (brass sheet for instance) but then neither do any of the mid priced lasers, for cutting metal you are looking to spend over 20 grand, now that is serious money in anyone's book!