Why is DNA double-stranded, but RNA single-stranded?

When we refer to DNA as being 'double-stranded', we mean that it consists of two strands of DNA bound together. These strands can be separated from each other either temporarily (e.g. when transcribing a gene into RNA) or permanently (e.g. when duplicating genetic material prior to mitosis), and each can act as a template for the synthesis of a complementary nucleic acid (either DNA or RNA), using the original strand as a template.

So why isn't RNA double-stranded?

Actually, a better question is to ask why DNA is double stranded in the first place. Many people assume that being double-stranded allows for it to be copied, but that explanation doesn't hold water. It could just as well be single-stranded - both RNA and DNA are only copied from a single strand of DNA anyway. In fact, it would save on the whole 'unzipping' process that DNA has to undergo just to present a single strand to RNA or DNA polymerase*.

No, a better answer comes from understanding that DNA is absolutely frantic in its efforts to keep itself from undergoing mutations. Mutations are generally bad things (Dawkins once famously said that there are many more ways of being dead than they are of being alive), and a zero-mutation rate is evolutionarily favoured. DNA, not RNA, is the ultimate genetic repository of information, and so you would expect it to be fiercely guarded. In this regard, being double-stranded helps in at least two ways.

Firstly, the 'information' part of DNA is the nitrogenous base, as opposed to the pentose sugar or the phosphate residues. In a single-stranded molecule, this important part would be exposed to the cellular environment, providing more opportunity for it to be mutated by the various chemicals there. In a double-stranded configuration, however, the two nitrogenous bases are locked within the complex, facing each other in the centre of the molecule. This organisation helps to safeguard them from local mutagens.

Secondly, having two complementary strands facing each other fundamentally means having two copies of the same thing placed right next to each other. This allows for proof-reading. George C. Williams summarised this beautifully in this pithy passage** (recall that adenine [A] on one strand should always bind to a thymine [T] on the complementary strand, and visa versa; likewise cytosine [C] always binds to guanine [G], and vice versa):

[Imagine a] gene containing the sequence CCAXT. The X cannot possibly be right because, whatever it is, it is not one of the expected DNA components. Fortunately, all we need to do here is to consult the complementary strand of the DNA. If it is GGTCA, we immediately know that the X should be replaced by a G.

In this way, mutations may be corrected or at least limited. (Actually, the proof-reading exercise and mechanism is not usually so simple as the above, but that doesn't matter for the present purposes.) And the proof is in the pudding. When you compare viruses that use DNA as the repository of their genetic material with those that use RNA, it can be seen that RNA has a higher mutation rate than the more robust and correctable DNA.

When used mainly as an intermediary between genes and proteins (rather than as an ultimate repository, like with the viruses above), RNA doesn't need this elaborate double-stranded structure. It is rather unlikely that the mRNA strand will mutate before it carries out its job anyway, but if it should, a perfect new one could always be synthesised from the DNA again. More importantly, the chemical inertness that double-strandedness grants to DNA would be a hindrance in the case of RNA, which relies on its ability to fold and contort somewhat (a little like a protein) to accomplish some of its tasks.

* the enzymes that make RNA or DNA from a single strand of 'parent' DNA, respectively

** The quote is from Williams' excellent book, "The Pony Fish's Glow" (1997)