When you need to control an appliance with an Arduino output, a relay actuated by the Arduino is a good intermediate interface. The coil of the relay will typically be low-voltage DC rated at 5 V. The relay contacts will typically be either one Form A, single-pole, single-throw, normally-open contact, or a Form C, single-pole, double-throw contact, meaning there is a moving contact that is connected to a normally-open contact when the relay is energized as well as that moving contact having a connection to a normally-closed contact when the relay is de-energized. If you want a circuit to close when the relay is energized, the normally-open contact and the moving contact are wired in series with the load. If you want a circuit to open when the relay is energized, the normally-closed contact and the moving contact are wired in series with the load.
The main advantage of the relay is the contacts are isolated and you are free to use them as an Arduino-controlled switch to turn loads on and off. There is a problem, however, when you use a relay with a mains-powered load such as a heating blanket, a TV, or lights in a room. This places dangerous 115 VAC power on the relay contacts thus presenting a shock hazard. If you are breadboarding circuits with the Arduinos, this potentially places 115 VAC power on your breadboard. One little mis-connection and poof! a whole bunch of stuff goes up in smoke. For your safety, it is best NOT to use a relay to switch mains power, especially the tiny "crystal sized" relays with pins spaced to allow their insertion on a solderless breadboard. There is a better, safer, way to switch 115 VAC power with the Arduino as described below.
Purchase a solid-state AC relay with the capability of handling the current you want to switch.
Here is a typical device you should consider, available from Wal-Mart for about US$11.00. This device is capable of switching up to 25 amperes of current, at AC voltages from 24 to 380 V AC. It will easily control your heating blanket, or a TV, or room lighting. This is a four-terminal device. Two of the terminals accept a DC control input, typically in the range of 3 to 32 V DC. The other two terminals are connected in series with a mains-powered load, exactly as you would wire any other two-terminal switch. You can drive the control input terminals directly from an Arduino output pin, paying attention to polarity. The (+) input pin would connect to an Arduino output; the (-) pin would connect to your Arduino common.
The device should be properly packaged in a metal box with a cartridge-fuse holder in series with the "hot" or black wire of a line cord, attached to the box with a strain-relief bushing, the other side of the fuse connecting to one of the two AC terminals on the solid-state AC relay. A three-prong convenience outlet should be mounted on the box for plugging in the load. Wire the "hot" terminal (dark or brass colored) of the outlet to the other AC terminal on the solid state AC switch. Wire the "neutral" or white terminal (silver colored) to the "neutral" or white wire in the power cord. Connect the green "safety ground" in the power cord to the metal box and to the "safety ground" on the convenience outlet. You can mount an RCA jack on the box to bring in the low-voltage DC control signal, or just drill a hole and install a rubber grommet to feed the control wires into the box and connect them directly to the DC control terminals of the solid-state AC relay.
Alternatively, you can operate the solid-state AC relay from a relay that the Arduino controls. Just connect +5 V DC power to the moving contact of the relay and connect the normally-open contact to the (+) input of the solid state AC relay. Connect the (-) input of the solid-state AC relay to common. Use low-voltage 24 AWG stranded twisted-pair wires to connect from the relay contacts (or the Arduino output) to the solid-state AC relay. You can also use speaker cable or zip cord cable, but that would be overkill for this application.
Wireless control is quite ambitious for a novice Arduino user. What shields are you planning to use for wireless communications?
you just have to look at a picture and do the same thing.
That statement scares the hell out of me. Looky, makey is okay for lower-order primates, but here on Electronics Point we expect more along the lines of learning and understanding before actually making. Simplifies the troubleshooting if it doesn't work right away.