How does investing in stocks online work?

They are called "shares." Brokerage fees are paid per transaction. If you buy 100 shares at once, you'll pay one brokerage commission. If you make four different purchases of 25 shares each, you'll pay four brokerage commissions.

Buy low, sell high.

Transactions fees are dependent entirely on how the brokerage charges you. Generally, most brokerages charge a flat fee for each trade (warning: a buy and a sell is considered two trades), like $7.95 (for a total of technically, $15.90).

As for how much the price can fluctuate, nobody can tell you this. You're literally asking whether a stock will be profitable. However, if you're wondering whether the stock's price may change after you set an order and prior to it being filled, that's generally dependent on the hours the market is open. The price can change in any fashion during open market hours (which is when you generally set your orders).

American stock markets are open during 9:30 AM to 4:00 PM. Pre-market and after market trades don't occur much, so price does not change during that time (it really only changes in special events, such as earnings announcements).

There are a lot of catches. You'll have to look them up. But generally, expect to lose everything if you have no idea what you're doing. And yes, you can go in debt, if your trades turn out to be absolutely awful.

There's no way to gauge profitability of a market. It's entirely how you trade.

Trading platforms are available from third-party companies, and also your own brokerage (usually). Most brokerages allow alerts to be set.