The serial port is a type of connection on PCs that is used for peripherals such as mice, gaming controllers, modems, and older printers. It is sometimes called a COM port or an RS-232 port, which is its technical name. If that's not enough to confuse you, there are two types of serial ports -- DB9 and DB25. DB9 is a 9-pin connection, and DB25 is, you guessed it, a 25-pin connection.
A serial port can only transmit one bit of data at a time, whereas a parallel port can transmit many bits at once. The serial port is typically the slowest port you'll find on a PC, if you find one at all. Most newer computers have replaced serial ports with much faster and more compatible USB ports.