Limit order: A limit order differs from a market order in that the trade is only completed at a certain price. For example, if you enter an order to buy 10 shares of Nike at $70 each, the order will only go through if the broker can fill at it at a price of $70 per share. Limit orders are a good way to buy and sell stocks that trade less frequently, since there may not be enough willing sellers to fill a market order at a reasonable price. They are also good for stocks that you feel are too expensive right now, but that you'd be willing to buy if the price dropped. These orders are a good for "set and forget" investing, since you can place a limit order that will remain in effect until a stock reaches the price at which you'd like to buy.
Online discount brokers: This label is generally given to the companies you see on the list here. While discount brokers are increasingly offering "extras" like research on stocks and funds, they primarily exist to help you place orders to buy investments at a very low cost. Many investors don't need the hand-holding of a full-service broker, and would prefer to save money by paying no commission for online stock trades. That way they ensure more of their money goes toward their investment portfolio, not paying for frills.
Market order: This is an order that will be placed immediately at the prevailing market price. Thus, if you enter an order to buy 10 shares of Amazon, your trade will be filled by matching it with someone who wants to sell shares of Amazon, though not at a known price per share. I like to call this the "get me in!" order type, since it will be filled quickly, although you could end up paying a slight premium. In most cases, however, you'll end up paying the stock's "ask" price, or very close to it.
A broker is simply a licensed person through whom you can buy and sell these stocks. When you use an online platform, it essentially acts as an online broker. If someone wants to buy any product, he can contact a brokerage service. They would place an order and let him know the market price of the product. If the investor is interested, they would include a commission for themselves and process the order forward. If the user wishes to buy stock, the broker would forward his order to a stock exchange, which would approximately take 3 days to complete including the money exchange between the broker and the investor.