A partially filled order is n order for which a portion of the request to purchase or sell has been executed, but not all.

For example: 

You place a BUY order to BUY 10 coins @ R100 per coin. A trader comes a long and deposits 10 coins, places a SELL order for 1 coin @ R100 per coin and another sell order for 9 coins @ R110 per coin.

Your order will now be partially filled. You will have 1 coin balance and another 9 coins outstanding on your partially open order (you only need 9 more coins to fill the complete order). The other trader will now be left with a closed order for 1 coin @ R100

PRO-TIP: You can create, cancel and accept as many orders as you want, play around with the numbers to make your orders more attractive to other traders.