An "option" is a generic name for a contract that gives you the option of doing something in the future. The stock options employees get as part of their compensation and the options stock traders deal in (a financial instrument) are very specific forms of those kinds of contracts.

The "options as a compensation mechanism" are primarily due to the tax efficiency of options as a whole.

The options contract market is massive, in the public markets they are fungible and actively tradeable. In private equity markets they are mostly limited to compensation mechanisms.

Of course, they are a huge market. You can do it with most brokers including robinhood.

Not only can you buy options, you can also write the contract and sell it on the market.

Options are probably the most common mechanism for a person to leverage themselves. An out-of-the-money option will yield far more than simply buying a single share. That said, it's a lot more risky.

There's loads of different options. A quick introduction:

https://www.investopedia.com/options-basics-tutorial-4583012

This is what "kids" on Robinhood were doing the last year or so.

Yes, you can purchase (and sell) options via many brokerage websites. Robinhood became quite popular because it made it very easy via an app for the masses.

You can trade options with just a couple of bucks or on margin. You don't need a bunch of cash. I've done it on Robinhood and bought some calls for a few hundred bucks.