Pending market orders, where are they stored while awaiting execution?

 
 Stop loss, trailing stops, and take profit, are they executed by my computer or from an external server? If my computer is powered down, will the order be executed?
 
If you are using your computer and the terminal running off of your computer, then it needs to be running to operate trades.  There is the possibility of setting up and external terminal running on a VPS that is specifically designed to be running all the time.
 
  1. There are pending orders, sent to the server, and become market order when market reaches price
  2. There are market orders, sent to the server and filled.
  3. There are no pending market orders.
 
Sethh:
 Stop loss, trailing stops, and take profit, are they executed by my computer or from an external server? If my computer is powered down, will the order be executed?

If  a stoploss and/or takeprofit have been sent to the broker and appear in your trade window, they will be executed whether you are connected to the broker's server or not.

Trailing stops are calculated on your computer, if connection to the broker is lost, it will stop trailing.