In traditional payment systems there is always someone "in charge" — a bank, a processor, a government. Bitcoin has no one in charge: every participant holds the rules, and no one can rewrite them.
Transactions are packed into a "block" — roughly every 10 minutes. Inside: a list of transfers and a signature.
Every block references the previous block's hash. Changing an old block = recomputing the whole chain.
All nodes hold the same copy. Any tampering attempt is rejected by the majority.
Mining is a lottery. Computers worldwide try trillions of numbers per second so that the new block's hash starts with the required number of zeros. Whoever guesses first adds the block and gets the reward.
The more computers join, the harder the puzzle. The network auto-adjusts difficulty so a block is found roughly every 10 minutes.
The public one — share it freely to receive coins. The private one — guard it: whoever knows it owns the coins.
The address where Bitcoin is sent to you. Post it on a website, your profile, a coffee shop wall — fine.
256 random bits — more combinations than atoms in the visible universe. Lose it — lose the coins. Show it — give away the coins.