I'm looking to buy shares through a stockbroker, but before I give them the order to purchase shares from the company I want to invest in, I want to check beforehand if that company is listed. How can I check this?
A company whose stock is available for sale to the public is called a publicly-held or publicly-traded company. A public company's stock is sold on a stock exchange, and anyone with money can buy shares through a stock broker.
This contrasts with a privately-held company, in which the shares are not traded on a stock exchange. In order to invest in a private company, you would need to talk directly to the current owners of the company.
Finding out if a company is public or private is fairly easy. One way to check this is to look at the Wikipedia page for the company. For example, if you take a look at the Apple page, on the right sidebar you'll see "Type: Public", followed by the stock exchange ticker symbol "AAPL". Compare this to the page for Mars, Inc.; on that page, you'll see "Type: Private", and no stock ticker symbol listed.
Another way to tell: If you can find a quote for a share price on a financial site (such as Google Finance or Yahoo Finance), you can buy the stock. You won't find a stock price for Mars, Inc. anywhere, because the stock is not publicly traded.
Hmm... Well there are several ways to do that:
Go to any bank (or at the very least major ones). They can assist you with buying and/or selling stocks/shares of any company on the financial market. They keep your shares safe at the bank and take care of them. The downside is that they will calculate fees for every single thing they do with your money or shares or whatever.
Go to any Financial broker/trader that deals with the stock market. Open an account and tell them to buy shares from company "X" and keep them. Meaning they won't trade with them if this is what you want.
Do the same as point 2, but on your own. Find a suitable broker with decent transaction fees, open an account, find the company's stock code and purchase the stocks via the platform the broker uses.