How do I turn on a switch without being at the switch?
What you need is a relay. A relay is an electromechanical switch that is activated by electricity and uses magnetism to turn the switch either on or off. The purpose of a relay is to switch high power items that can draw too much current to safely come in contact with the switch itself, or to enable you to run high currents through one wire and connect and disconnect them with low current through another wire. Basically, a relay is a switch that keeps you away from dangerous amounts of power, or lets you use smaller wires to run things.
How does it do that?
A relay has two main parts: the contacts and the electromagnet. The contacts work exactly the same way as a mechanical switch's contacts. You attach the incoming wire to one contact, and the outgoing wire to the other contact. The switch either connects the two contacts, allowing electricity to pass through, or it disconnects them, breaking the path from the power source to the ground and stopping the power in its tracks. The electromagnet in a relay takes the place of human fingers. When current is run through the relay's electromagnet (from a different circuit than the one you are turning on and off with the relay), energy is forced through a coil of copper wire. This energy creates a magnetic pull on one of the contacts, physically bending it until it touches the other contact, creating a closed circuit. When the current stops flowing through the coil, the magnetism disappears and the contact restores to its original position, no longer touching the other contact.
Where does this magnetism come from? I don't see any magnets!
As Maxwell and Faraday deduced in the mid 1800's, when electricity runs through a coil of wire, a proportional magnetic field is generated around the coil of wire in a perpendicular direction to the coil. Basically, the coil of wire converts electrical energy to magnetic energy, instantaneously creating a magnetic field.
Why do I need a relay?
You don't. If you are supplying power to a small electric motor through thin 24 gauge wire, you need nothing more than a battery, wire, motor, and a switch. The motor will draw at most a half amp of electricity, which will pass through the thin wire just fine. If you upgrade the motor to something larger, you will draw more amps to provide the added boost, and therefore need a larger wire. Think of people in a hallway. If too many people try to get through at the same time, the hallway gets clogged. And if they stay there for a while, because they are moving more slowly, the room can get a little warmer from everyone's body heat. The same is true for electricity. If too much electricity tries to go through a wire that is too small, the electricity backs up, and the wire heats up. As a result, with the larger motors, you need a larger wire.
Once you need larger wires, you will need larger switches that can handle the added power. And if you have several motors, you will need several large switches and several less flexible larger wires. All of this begins to become too big and unwieldy to be very useful, so it's a good idea to use relays to change the way things work. With relays, you can still use small wires and small switches to turn the relays on and off. And you can run just one pair of big wires to the relays, which can split up the one wire into several short lengths of large wire that go between the relays and the motors. This will make the control box a more manageable size and the cabling going between the control box and the motors more flexible and easier to work with.