This topic came up on an email list I'm on, so I thought I'd write up this little reliving of high-school organic chemistry as a blog post.
The virtue of carbon in particular is that stretch and bend can be very carefully controlled. Carbon atoms have four very strong bonding sites, which enable it to bond together into very controllable structures. You can make strings, sheets, or three-dimensional solids. You may have heard of these three-dimensional highly pure carbon lattices; they're called diamonds. That should give a sense of how strong carbon bonds are.
But for things like bicycle building, you can get your carbon atoms to bond together into a flat surface, and then you layer a bunch of these flat surfaces together, and it works like a truck's leaf spring: You have a very strong material and you can control the way that it will flex and bend by the way that you orient the fibers. (Laterally stiff yet vertically compliant, yes!)
And if you get more-or-less randomly oriented fibers, then they'll be in microscopic sheets bonded together by occasional cross-sheet single bonds--and the sheets will easily slide off. A great example of this is the graphite in pencils; it easily sheers off under friction into a grey powder--and the sticks of graphite break easily, where a piece of metal of the same thickness would bend.
If you look at the periodic table of the elements, you'll note that Silicon lies right under carbon. It has the same orientation of four bonding sites but for various reasons, those four bonds are less flexible than carbon's--you basically can't make silicon fibers like you can with carbon. But the properties of high-purity solid silicon--better known as glass--are familiar to us. Solid glass has a pretty random assortment of bonds, like pencil graphite, but most of them will be three-dimensional bonds (when you increase the number of three-d bonds, you make crystal glass), making glass a very strong material (stronger than wood, in fact). That's why you can build very tough things like boat hulls from glass fibers embedded in a flexible matrix.
That's the usual failure mode of carbon fiber as well; once it goes beyond (or in a different direction from) its designed flex, the whole thing can just explode (in a better scenario, some of the layers go and you have enough warning to stop the use before the whole object gives way.
But when it breaks, it breaks catastrophically: if some atom-to-atom bonds start breaking, the rest of the material isn't flexible enough to bend; the whole thing just gives way. Similarly, when you look at broken metal pieces, they're either twisted and stretched or they've parted suddenly and you can see the crystalline structures inside them, due to either bad metallurgy, or (if I understand correctly) in the case of alloys, flexing.
If you want to understand metallurgy, get your hands on a stick of butter. First, break it in half and note how it crumbles at the breaking point. Then knead it for a minute: you can use a knife in a dish, or you can do what pastry chefs do and work it under cold water (there's a reason you use ice water and butter to make pie crusts!). You'll find that the material becomes much more flexible and stretchy, much less likely to break and crumble. This is what you want if you're making puff pastry (very thin layers of butter and flour) and also what you want your metal to be like if you're making something like wire, that you want to be flexible and strong, or a bike frame. The real genius of metallurgy is making things that will be strong and tough and springy--something that butyrology has yet to master!
The Other Big Problem with composites, in my book, is their irrecyclability. The epoxy matrix in which they're embedded is very tough stuff, but it can't be melted down because it's made of big complex molecules; they have to break down chemically, and it's not practically feasible at this point to reverse the chemical reactions that it took to make epoxy. (If you've dealt with epoxy, you know hoe much heat it gives off as it cures; all that energy would have to go back into the epoxy in order to break it down, and heat alone doesn't do it.)
Once composites start getting soft; they can't be melted down and re-used but they also can't be trusted with your life and safety. That's the reason why many (most?) dumps have an express prohibition on boat hulls of any kind.