What is a stock?
A share of stock represents ownership of a portion of a corporation. In olden times, you would get a physical stock certificate (looking something like this) with your name and the number of shares on it. That certificate was the document demonstrating your ownership. Today, physical stock certificates are quite uncommon (to the point that a number of companies don't issue them anymore). While a one-share certificate can be a neat memento, certificates are a pain for investors, as they have to be stored safely and you'd have to go through a whole annoying process to redeem them when you wanted to sell your investment.
Now, you'll usually hold stock through a brokerage account, and your holdings will just be records in a database somewhere. You'll pick a broker (more on that in the next question), instruct them to buy something, and they'll keep track of it in your account.
Where do I get a stock?
You'll generally choose a broker and open an account. You can read reviews to compare different brokerages in your country, as they'll have different fees and pricing. You can also make sure the brokerage firm you choose is in good standing with the financial regulators in your country, though one from a major national bank won't be unsafe.
You will be required to provide personal information, as you are opening a financial account. The information should be similar to that required to open a bank account. You'll also need to get your money in and out of the account, so you'll likely set up a bank transfer. It may be possible to request a paper stock certificate, but don't be surprised if you're told this is unavailable. If you do get a paper certificate, you'll have to deal with considerably more hassle and delay if you want to sell later.
Brokers charge a commission, which is a fee per trade. Let's say the commission is $10/trade. If you buy 5 shares of Google at $739/share, you'd pay $739 * 5 + $10 = $3705 and wind up with $3695 worth of stock in your account. You'd pay the same commission when you sell the stock.
Can anyone buy/own/use a stock?
Pretty much. A brokerage is going to require that you be a legal adult to maintain an account with them. There are generally ways in which a parent can open an account on behalf of an underage child though. There can be different types of restrictions when it comes to investing in companies that are not publicly held, but that's not something you need to worry about. Stocks available on the public stock market are available to, well, the public.
How are stocks taxed?
Taxes differ from country to country, but as a general rule, you do have to provide the tax authorities with sufficient information to determine what you owe. This means figuring out how much you purchased the stock for and comparing that with how much you sold it for to determine your gain or loss. In the US (and I suspect in many other countries), your brokerage will produce an annual report with at least some of this information and send it to the tax authorities and you. You or someone you hire to do your taxes will use that report to compute the amount of tax owed.
Your brokerage will generally keep track of your "cost basis" (how much you bought it for) for you, though it's a good idea to keep records. If you refuse to tell the government your cost basis, they can always assume it's $0, and then you'll pay more tax than you owe.
Finding the cost basis for old investments can be difficult many years later if the records are lost. If you can determine when the stock was purchased, even approximately, it's possible to look back at historical price data to determine the cost.
If your stock pays a dividend (a certain amount of money per-share that a company may pay out of its profits to its investors), you'll generally need to pay tax on that income. In the US, the tax rate on dividends may be the same or less than the tax rate on normal wage income depending on how long you've held the investment and other rules.
