In short, your Bitcoin address is a Public Key. You alone hold your Private Key and you must keep it safe like a bank password. When you send money, your Wallet takes data representing how much you're sending and who you're sending it to and creates a Signature to accompany it. The Miners will hold this transaction until the next Block is mined. Once the block is mined, the transaction becomes part of the public Blockchain for all eternity. All computers across the world a part of the Blockchain will recognize this transaction and the adjusted values of both wallets involved.
Links to websites to buy and trade Bitcoin are at the bottom of the post.
Bitcoin is a decentralized currency whose value is backed by its proof-of-work procurement.
Modern currencies are fiat, meaning that the value of the money comes from the faith that the government representing them is stable and can back the value of it.
Unlike modern currency, no one company or person controls Bitcoin. Instead, many millions of computers record the ledger of every transaction ever taken place through Bitcoin. This is called the Blockchain, and anyone may read it. When people talk about the Blockchain, they are referring to this public ledger that every member of the Bitcoin ecosystem keeps track of.
The computers pay close attention to the Blockchain because this is the proof that a wallet has money to spend when it attempts to make a transaction. The computers check to see if that wallet has the value it claims to have by looking to see if money has entered it in the past. If the computers cannot find the money this wallet wants to spend, it declines the transaction.
When you send money by Bitcoin, the program that keeps track of your money (your wallet) will look to see if multiple nodes confirm the transactions. It usually wants at least 6 confirmations from different nodes before believing the transaction was valid. This is what is meant by "decentralized". No one bank authorizes a transaction, instead many millions of computers will use the Blockchain records to give Confirmations on a transaction.
Your wallet creates transactions by using your private key to sign a transaction attempt. This is the crypto part of cryptocurrencies. This is very complicated to explain but is important to all modern computing. Keys in computing are long strings of gibberish characters, very unique and almost impossible to guess. They can be used to sign anything you want by taking a file or a block of text and running it through a process to generate a signature. Signatures are a special combination of your private key's information and what you are signing. Your private key also has a public key counterpart that can safely be shared. People with both your public key and a signature you created can use a program to independently verify that your private key created that signature, and that the signature represents the associated data.
When you send or receive money, it will be held for verification until the next Block is mined. Blocks are universally predictable cryptographical problems created for computers to solve by guessing thousands of times a second. The better a computer is, the more it can guess the answer to this problem. Globally there are millions of computers guessing billions of times a second. Many miners put their computers together in what's called a mining pool, then split the bounty for the problem when it is solved to create more predictable rewards for their efforts. Miners are paid not only the value of the Block, but the value of all transaction fees from all pending transactions waiting on that Block to be solved.
This process is what builds the Blockchain, and the rewards for mining halve predictably. It was initially 50 BTC per block, now it is 12.5, and will be 6.25 in about 2 years as of writing.
If you are worried that these keys are vulnerable, keep in mind your bank uses a version of private/public key signing for their websites. Those green "Secure" locks in the URL bar represent that there is an SSL certificate encrypting traffic. SSL is another type of cryptographic signing almost identical to how private / public keys work for Bitcoin. When you sign a transaction with the Blockchain, it is the very similar with how your bank signs the data on a webpage and sends it to you.
If you're worried these keys can be guessed, be aware there are currently more possible Bitcoin addresses than there are atoms of matter in our galaxy. If you divided the Pacific Ocean into milliliters of water, you'd still be dealing with a number about a trillionth the size of the number of Bitcoin addresses. It is so much more likely your existing bank account passwords could be guessed than there could ever be two generated private keys that generate the same bitcoin wallet.
These platforms use existing banking to buy bitcoin. I don't trust them, but they're good for normies.
This platform lets you use credit cards to buy Bitcoin, but have high fees.
LocalBitcoins lets you use direct money transfers for Western Union, Zelle, Venmo, or even meet up in person in many cities all over the world to buy in person.
Electrum is the oldest and most trusted computer wallet, completely open source. With Electrum, you can write your pneumonic phrase on a piece of paper, put it in a vault, and completely delete Electrum off your computer until you need it.
Jaxx is a convenient multi-currency wallet, but it's not entirely open source.
Mycelium is a good phone wallet.