Market buy and market sell orders are the two sides of the market order, which is covered in full under that entry, including how such orders are filled and when they are used. This page explains the two names as traders meet them on an exchange screen.
A market buy tells an exchange or broker to purchase a set amount of an asset straight away at the best prices currently offered by sellers. A market sell does the reverse, selling immediately to the highest bids in the order book. The trader gets speed and certainty that the order will fill, but not certainty about the final price.
That price uncertainty is the main thing to understand. A large market order can work through several price levels in the book, so the average price paid ends up worse than the price shown when the order was placed, a cost known as slippage. The effect is small in deep markets for major coins and can be severe for thinly traded tokens or during fast moves. Because market orders take liquidity rather than add it, they are charged the taker fee on most exchanges, as described under market taker. Traders who want to control their price instead use a limit buy or limit sell, accepting that it may not fill.