Market-On-Open Order (MOO)
  
A market order simply means that a trader wants to buy a stock at whatever the current retail price is for that stock.
Google a stock price...the number you see provides the market price. That figure represents the culmination of all the bids and asks that take place under the surface.
Put in a market order and you aren't specifying a price you want to target (targeting a specific price is called a "limit order"). You'll just take whatever it's selling for.
Meanwhile, the "On-Open" part here refers to the open of the market. You know that morning bell they sometimes ring on TV? That's the open. Because yeah...as much Red Bull and ephedrine a stock trader can ingest, they have to sleep sometimes.
So the New York stock markets close at 4 pm EST each work day, and open up again at 9:30 am EST the next work day. That 9:30 time marks the open.
Now, despite the pretense that the market is "closed," some trading takes place overnight. Robots, insomniacs, and people living in other time zones keep up a (relatively) thin trade in the after-market and pre-market periods.
Because of that extended-hours trading, the opening price for a given stock might not match the closing price recorded the day before.
A market-on-open order means that a trader will buy a set number of shares at whatever price the stock opens at on the next trading session. So, on Monday night, a trader might set a MOO to purchase 100 shares of a stock that closed the previous session at $50. The stock opens on $52...so the MOO order executes, buying the trader 100 shares at $52 a share...the market price at the open.