If only a portion of your bids or asks can be matched with the current order book, the matched orders will be executed, and the remaining orders will stay on the order book.
For example, if you placed a bid for 2 shares at $10, but there is currently only 1 ask on the order book for $10 (and no other asks), then you would receive 1 share. Your other bid would stay on the order book until another user puts in an ask order of $10 or above.
The reverse is also true if you placed an ask, that is, if you put a bid for 2 asks at $10, but there is currently only 1 bid on the book for $10, you would only sell 1 share (for $10). Your other ask would remain pending on the order book.