A Market Order attempts to buy/sell at the current market price. It buys or sells “right now.” What it does is it buys or sells available limit orders sitting on the books right now. Thus, slippage may occur (where you get a slightly higher price with a buy market order or slightly lower price with a sell market order). In very volatile times, slippage can be substantial. You’ll pay a fee for a market order as a rule of thumb. You can set a market buy or market sell.
A limit order places an order on the order book in hopes that it’ll be filled by someone else’s market order. When the market price reaches that price, it’ll buy or sell (AKA “fill”) if there is a buyer or seller. With the exact mechanics of exchanges aside, the basic concept here is that someone else is placing a market order and that a buy or sell will fill your limit order. Limit orders aren’t subject to slippage and sometimes have lower fees than market orders. You can set a limit buy or limit sell.
A stop order places a market order when a certain price condition is met. So it works like a limit order, in that it goes on the books, but it buys or sells like a market order. Stop orders are therefore subject to the same fees as market orders and are subject to slippage. You can set a stop buy or stop sell.
A Trailing stop sell order sets the stop price at a fixed amount below the market price with an attached “trailing” amount. As the market price rises, the stop price rises by the trail amount, but if the stock price falls, the stop loss price doesn't change, and a market order is submitted when the stop price is hit..