If you've lost the key to your push and twist door lock, or if the keyhole is damaged and you can't insert a key, don't despair. There are a few ways to open these locks without the key.
One way is to use a length of stiff wire, such as a coat hanger. Straighten out the wire and insert it into the keyway. Push and twist the wire until you feel it catch on something inside the lock. This will hopefully engage the locking mechanism and allow you to turn the knob or lever to open the door.
Another way is to remove the doorknob (or lever handle) itself. This will usually require a screwdriver. Once the knob is removed, there will be a small hole in the plate behind it. Insert a straightened paperclip or other thin object into this hole and push until you feel resistance. This should engage the locking mechanism so that you can turn it with your paperclip and open the door.
If neither of these methods work, your last resort is to call a locksmith who will be able to pick or drill your lock open without damaging it too much.