A charged object does not need to touch another charged object to push or pull on it. Around every charged object there is an electric field. An electric field is the region where an electric charge experiences a force.
If you place a second charge anywhere in this region, it feels a force. If the charge leaves the region, the force becomes too small to notice. The force is caused by the field, not by contact, so it is a non-contact force, like gravity or the force between magnets.
The direction of the force depends on the sign of the charges. Like charges repel and opposite charges attract. The field is strongest close to the charged object and gets weaker with distance from it.