I currently own two folding bikes, both made by Dahon, a company celebrating their 25th anniversary this year. I also work for a bicycle shop that specializes in folding bikes, along with recumbents and bikes for special needs. So I know a bit about folding bikes. In addition to the Dahons, I've ridden and worked on Bike Fridays, Birdys, Breezers, and finally got a chance to ride a Brompton a month or so ago. They all have their strengths and weaknesses, and all have their advocates and detractors. I'm not going to go into the pros and cons of any one brand or model, but simply want to talk a little about the virtues (and vices) of the breed as a whole.
So what are those virtues? Why the heck would anyone want to ride such a funny-looking bike with tiny wheels? Well, to start with, it's just plain fun. Huh? Yes, that's right, they are fun to ride. Those tiny wheels accelerate really quickly, and once you adjust to the different handling, steer nimbly around things. Yes, it's a different feeling, and for long distance, steady speed, larger wheels have an edge, but for bopping around town, running errands, commuting, these little guys are great. Another feature that I like for around town is the low frame, much like the classic "step through" frames that Americans tend to describe as "lady's bikes", but which much of the world sees as simply practical. Try it sometime, and see how nice it is to just step over the frame, not having to swing your leg up and over the seat. It's pretty handy, especially when carrying a load.
Now, the big thing that sets folders apart is... well... they fold. Different brands and models fold differently, but basically you get a fun-to-ride bike that you can collapse into a smaller package for transporting it by another means. This means you can store it more easily in an out of the way corner, and you can toss it in the trunk or back seat of your car for recreation, or shopping, or even as a sort of "life boat". Think about it... if your car breaks down... and you cell phone melts... and nobody's around to help.... okay, so that's a rare circumstance, but you get the picture. As for other forms of transportation, folders are ideal for going on buses, trains, planes, subways, trolleys, you name it. Some designs lend themselves more to one form of transport than others, so make sure you do your research and ask lots of questions. In general though, any folder will expand your options and add another level of versatility to your cycling life.
Give one a try, and I think you'll like it. Take it for a test ride, see how it handles, and give yourself enough time to get accustomed to the different handling. At first it might feel a little odd, but you'll adapt very quickly. Have the salesperson show you how it folds, several times if needed, and try folding it yourself. Pick it up, carry it around. Check out any carrying bag options. Then, while it's folded, just look at it for a while, and think of all the great places you can bring it and how easy it is to store, and I think you'll better understand the appeal.
(Pictured is my 2005 Dahon Speed P8, with minor modifications from stock, and liberally plastered with militant pro-cycling stickers. My other Dahon is the groovy 2006 limited edition single speed Hon Solo.)