The halving

Every 210,000 blocks

The block reward halves every 210,000 blocks. At 144 blocks a day that is roughly four years. It started in 2009 at 50 bitcoin per block.

210,000 divided by 144 is about 1458 days, so just under four years.

The series

50, then 25, then 12.5, then 6.25, then 3.125, and so on. Each step holds for 210,000 blocks. At some point the reward is smaller than one satoshi, and then there are no new coins at all.

Hence the 21 million

The sum of all rewards is 210,000 x (50 + 25 + 12.5 and so on). That series approaches a fixed value without ever exceeding it: just under 21 million. The cap is not an extra law, it is the result of the halving rule.

The cap follows from the halving, not the other way round.

What happens the day after

The yield per machine halves overnight, the electricity bill does not. Machines that were barely profitable no longer are and get switched off. That lowers the hashrate, whereupon the difficulty falls at the next adjustment.

What the halving does not do

It does not halve the price, nor the difficulty, nor the number of payments. It changes only how many new coins arise per block. What the price does afterwards, nobody knows in advance.

And in the long run

Once no new coins arise, miners live on fees alone. Whether those suffice to attract enough computing power is an open question, one of the few genuinely open ones in the design.