How do people earn Bitcoin?

There's no easy or foolproof way to make substantial amounts of bitcoin fast and "free bitcoins" don't exist.
Micro Earnings (pay to click websites, bitcoin faucets, Micro Jobs) Bitcoin faucets are a website or an app that function as reward system. They dispense satoshis (a hundredth of a millionth BTC) which can be claimed by completing a captcha or task.
Advertising/Writing about BitCoin (BitcoinTalk Signature Campaigns, writing for blogs and news-sites)
Providing bitcoin-related services (website manager, graphic design expert, etc...)
Bitcoin affiliate marketing

Bitcoin-trading app CoinBase
Day Trading: buying and selling bitcoin on the same day (depends on small, short-term price fluctuations).
CFD (contracts for difference): Similar to day trading but you don't have to actually "buy" the bitcoin, you just buy a contract for it (so you don't have to deal with holding or storing it). You can short sell for a profit.
Binary Options: predict whether the value of bitcoin will increase or decrease in a given time period. If your prediction is correct, you earn the payoff. If it's wrong, you lose your investment.

Some companies actually pay their employees in BitCoin and other cryptocurrencies

How are new BitCoins created?

Bitcoins are generated by a competitive and decentralized process called "mining".
"Miners" are process and verify transactions, securing the network with specialized hardware.
The first miner to verify the transaction receives a certain amount of bitcoins as a reward.
The Bitcoin protocol is designed in such a way that new bitcoins are created at a fixed rate.
This makes Bitcoin mining a very competitive business.
When more miners join the network, it becomes increasingly difficult to make a profit and miners must seek efficiency to cut their operating costs.

It requires A LOT of computing power, which it's pretty impractical to mine for yourself.

A block contains several transactions (2,400) and a special number.
Miners have to find a special number that when combined with the block, and put into a SHA256 method, it will give you a number with 60 0s in front.
The amount of 0s in front does not have to be 60, it depends.
Then it will broadcast the block to everyone else.
Then to reward the first miner, we will give them some rewards. That is mining.

Then why can we trust computation power?
As a rule, we assume the longest blockchain is the most updated one.
A person may try to forge one block or several blocks. But because creating a block necessitates finding that special number, it takes a lot of computation power to forge.
A person may be lucky enough to get a few blocks. But there are a bunch of other miners trying to create a block. So the forger cannot create blocks as fast as the rest of the world.
So after several blocks, there ends up being enough information to be sure which blockchain is the real one.

For the special number, it does not need to give you a hash number with 60 0s in front.
In fact, it changes periodically that will take around 10 minutes for miners to get that number.
When there are more and more miners, the number of 0s in front will increase. So the probability of you finding that number will decrease.
Why? Because the probability of finding a hash number with n 0s in front is 2^n.

Miners will get a fix number of bitcoin as reward. But this number decrease geometrically over time.
Right now the reward is 12.5 bitcoins.
This means the total number of bitcoin will never be over 21 millions.
Miners will also get a transaction fee that is left by the broadcaster because they want the miner to include their transactions into the block since each block has a limited transaction (2,400)