The actual number isn't 21 million.  People just round up to 21 million (and then say "the total amount will never be more than 21 million") when talking about it.

The actual limit on number of bitcoins that can be created is 20999999.9769

The number is calculated as follows:

Start with a block reward of a nice simple number of 50 BTC per block.

Then every 210000 block (approximately every 4 years if blocks are generated on average every 10 minutes), cut the block reward in half and throw away any decimals after the 8th decimal place.

So after 210000 blocks you will have generated 50 * 210000 = 10500000 BTC
Then after another 210000 blocks you will have generated another 25 * 210000 = 5250000 BTC for a total of 10500000 + 5250000 = 15750000 BTC
Then after another 210000 blocks you will have generated another 12.5 * 210000 = 2625000 BTC for a total of 10500000 + 5250000 + 2625000 = 18375000 BTC

If you continue this process, you'll find that by the time you reach block number 6930000 you will be cutting the block reward in half from 0.00000001 which, upon throwing away any decimals after the 8th decimal place, will be 0.00000000 BTC.  By then you will have generated a total of exactly 20999999.9769 BTC.  Since the block reward is then 0, cutting it in half leaves it at zero forever meaning that no new bitcoins will ever be generated.

You can see all the related numbers here if your curious:
https://en.bitcoin.it/wiki/Controlled_supply#Projected_Bitcoins_Long_Term

I think BIP42 is also relevant here.  In fact, after having read through that BIP, I can't tell if it's been implemented or not.  I also can't tell if the author is merely pointing out a quirk in the C++ code which formed the reference standard or if he's referring to Satoshi's intentions that the supply not be finite.

In any case, I'm sure DannyHamilton can enlighten us on it.

Thanks in advance!

Yeah, it's implemented.  They just check the number of doublings against 64 and set it to zero after that.

[edit]
To slow by me.

still none of you guys have said WHY, he asked why he choose those numbers not how it work

Well *why* may very much be a matter of speculation.  Who can really say "why" except Satoshi himself?  And if he hasn't said *why* then how can we infer it?  Seems like the next best thing to why is to look at what.  Anyway, in this case, I understood DannyHamilton's answer to be suggesting that the why was that the 21M was merely an artifact of the how.  Ie, there was no intentionality in the why 21M, that the intentionality was to have the supply diminish on a curve and 21M just happened to be the value for the particular parameters he chose---perhaps suggesting that if there were numbers chosen intentionally it was the 50BTC block reward and the 10 minute block time, and the halving rate.

Actually yes, Satoshi was very much aware of numeric representation issues.  And the fact is that you can't actually get 100% accurate rounding on basic-4 operations (addition/subtraction/multiplication/division) if you use more than *HALF* of the number of distinct values available.  So the actual limit on the number of divisions for accounting without "gotcha" rounding errors is 2^51, not 2^52.

Am I missing something here?  It appears to me that binary64 (IEEE 754) can precisely represent any integer from -2⁵³ to 2⁵³.  Even after allowing one bit to avoid rounding oddities we have a second bit unaccounted for.

The bitcoin.SE is a good reference on this:

http://bitcoin.stackexchange.com/questions/8439/why-was-21-million-picked-as-the-number-of-bitcoins-to-be-created

It's a good thing Satoshi didn't go much higher, because 21 * 10^14 is breaching on what a signed 64 bit integer can hold.

A signed 64-bit integer will go up to about 9.2 * 10¹⁸, much larger than the 2.1 * 10¹⁵ bound.