Nought is the more colloquial term for the figure zero or the number nothing, though both are widely used, except in the US, where zero is the only common one.

Well, that's the theory. When I started looking at where I actually say nought and where I say other things, I found it was quite mixed. The following are the observations of a British speaker, for whom "nought" is usually the most natural word. I believe they are generally applicable for most non-American forms of English, though as always individual speakers' judgements and usages vary.

Nought, zero, and oh

As the name of the figure, all three of these words are used, zero being the most formal and oh being the least formal.

0.2 is said "nought point two", while 0.02 is said "nought point nought two". I don't think we say "oh point...", but we can leave off the leading zero and write it .02 and say it "point oh two". We could more formally use zero instead of nought or oh in any of these places.

When someone asks how many there are, and there are none, you say "none" of course, but if you were going to name it emphatically as a number, you'd say "nought".

In telephone numbers it's always "oh". The London area code of 020 is always said "oh two oh" - I don't think I could use zero or nought there.

To curb hyperinflation you might cut three noughts off the currency. You inflate your budget tenfold by adding a nought on the end. That is, the nought is the actual figure "". These can also be called zeroes.

So the game of noughts and crosses (elsewhere called Tic-Tac-Toe) is always named with "nought", never zero or oh.

A car can accelerate from nought to sixty in such-and-such a time. Neither of the others is said.

Because nought sounds like eight it's not used over official broadcasts, like police and Seaspeak: so army operations are codenamed things like Bravo Two-Zero.

Hawaii Five-O has "oh".

We talk about the zeroes of a function, like the Riemann function. (Note on plural: British usually zeroes, American usually zeros.)

But the infinite cardinality is aleph nought, not aleph zero. (The American term for this is aleph null.)

We only ever say the zeroth power of a variable, or the Zeroth Law of Thermodynamics: we can't say nought-th or oh-th.

But in an polynomial equation we don't always add the "th" or the word "power". So, just as x4 is often spoken "x to the four", not fourth, so I think I'd say x0 as "x to the nought". (Never "x to the oh".)

Dialectally, the word ought is also used, but I'm not familiar with the dialects that say it, so I'll comment no more.

Nought, naught, and nowt

There's the very similar word naught, with an A, which also means "nothing". Obviously with two words with the same pronunciation and almost identical spelling and meaning, the spellings get confused, but here's the difference as used in books. These days, nought-with-an-O means nothing the number, whereas naught-with-an-A means nothing the amount of effect or result.

"It all came to naught." -- "All that work was for naught." -- "Set at naught." -- "Naught availeth." -- It's quite an archaic word really. But we could also write "set at nought", I suppose. (Fowler writes that it's "a matter of convenience only, & by no means universally recognized.")

Shakespeare makes a pun in Richard III: the wicked Richard of Gloucester is teasing the gaoler Sir Robert Brakenbury about the King's mistress, Jane Shore:

Glo. We say that Shore's wife hath a pretty foot,
A cherry lip, a bonny eye, a passing pleasing tongue;
...
How say you sir? Can you deny all this?
Brak. With this, my lord, myself have nought to do.
Glo. Nought to do with Mistress Shore! I tell thee, fellow,
He that doth naught with her, excepting one,
Were best to do it secretly alone.
Brak. What one, my lord?
Glo. Her husband, knave! Wouldst thou betray me?
Here Richard puns on "having nothing to do with" and "doing naughty things with". (The modern spelling switches from O to A but I can't be certain Shakespeare wrote it that consistently.) -- And the bit about betraying is because they both know the King is shagging her but can't say it publicly.

Although nought-meaning-zero is rarely used in America, my dictionary (and other nodes here) tells me it tends to be written with an A when it is.

In England there's a dialectical form "nowt" (rhymes with "out"), widely understood, meaning "nothing" or "none" or "no-one", as in the common expression of resigned amazement, "There's nowt so queer as folk".

The word naughty actually comes from naught, via an intermediate sense of "worthless". Ultimately they're all the same word, and the Old English is ná-wiht, that is "no wight". That's wight as in Tolkien's barrow-wights, a wight being an obsolete word for a creature.

Here endeth my brief history of nothing.