Serial means one event at a time. It is usually contrasted with parallel, meaning more than one event happening at a time. In data transmission, the techniques of time division and space division are used, where time separates the transmission of individual bits of information sent serially and space (on multiple lines or paths) can be used to have multiple bits sent in parallel.
In the context of computer hardware and data transmission: serial connection, operation, and media usually indicate a simpler, slower operation and parallel indicates a faster operation. This indication doesn't always hold since a serial medium (for example, fiber optic cable) can be much faster than a slower medium that carries multiple signals in parallel.
On your PC, the printer is usually attached through a parallel interface and cable so that it will print faster. Your keyboard and mouse are one-way devices that only require a serial interface and line. Inside your computer, much of its circuitry supports bits being moved around in parallel.
Your computer modem uses one of your PC's serial connections or COM ports. Serial communication between your PC and the modem and other serial devices adheres to the RS-232C standard.
Conventional computers and their programs operate in a serial manner, with the computer reading a program and performing its instructions one after the other. However, some of today's computers have multiple processors and can perform instructions in parallel.