This is the remains of a landline telephone junction box. It is missing the front cover, and would have looked similar to the picture in Machavity's answer.
There is a small chance that this is still live. You can test this by checking the voltage by using a multimeter. Take the 2 leads of the volt meter and connect them to the 2 wires coming out of the wall. In telecommunications jargon, these wires are known as the ring and tip. The green wire is the tip, and the red is the ring. The tip and ring corresponds to the old style phone jacks that telephone operators used in manual switchboards. They are similar to modern headphone jacks, but were mono.
It is fairly safe to leave this as is, but since you are remodeling, it would be a good idea to either replace it with new wiring even if you don't plan on using it. The wire you should use is known as Category (Cat) 3 wire. It is fairly inexpensive, and can be bought in bulk from home improvement stores. You will also need a telephone jack. Disconnect the wire from the old jack, and securely tie a piece of string to it. Then see if you can pull the wire from the other side. It may not be attached to the framing inside of the wall, so it should just pull out. You can use the string to pull the new wire through the original hole, so you don't have to do any drilling. Cat 3 wire has 4 wires, but you only will need to connect the red and green wires to the jack. This should be clearly marked on the jack. The rest of the wires won't be used. You can simply fold them back inside of the jack.
If the other end of the wire is not connected, you can just coil up the end of the wire and wrap it with electrical tape to keep it out of the way. That way, it would be easy to hook it up in the future. If it is connected at the other end, hook up the new wire exactly the same way as the old.