Wire, in reference to R/C, is the means by electrically connecting batteries to speed controls, speed controls to motors, servos to receivers and so forth.
Wire gauge, or "thickness," is very important in electric R/C's. Smaller gauge wire is fine for connecting certain low-current components such as servos, can be heated with a smaller soldering iron, is generally cheaper but has more resistance. Like the difference between a garden hose and a fire hose, the larger the wire, the more electricity can flow through it. Large gauge, high-quality wire used to connect high-current batteries and motors gives less resistance, which makes the vehicle faster. In some cases, large gauge is also slighly less flexible and some connectors are difficult to use with the bigger sizes of wire. Low-resistance, silicone-jacketed wire specifically for R/C use is highly recommended.
Gauge is represented by a number. The smaller the number, the thicker the wire. 12 gauge is the biggest used for high-current R/C applications; 18 gauge is the smallest.