A partially filled order is an order where only a part of the buy or sell order has been completed, not all.

For example:

You set up a BUY order where you buy 10 coins per 100 EUR / piece. A trader will come and set up a SELL order, where he would give 1 coin for 100 euros and 9 coins for 110 euros per piece. In this case your order is partially completed. 

You now have 1 coin and 9 waiting in open position to be completed. The other dealer closed with one coin / 100 EUR.