serial transmission works by sending data bit by bit over the same wire.. parallel transmission sends a few bits at the same time thorugh different wires.
generally serial ports require some additional hardware or software flow control.
parallel connection are generally cheaper.... and they seem to have more bandwidth (due to more wires)
serial connections make up for speed by better quality cables. (it's harder to improve quality for parallel wires).
typical applications (parallel):
the 25 pin parallel port on computers for old printers [This is probably the application for 'the' parallel port]
old IDE/PATA hardrives/ cd-roms connections to the mother board.. (still quite popular, being phased out though)
Most things are serial these days.. and it has been a trend for a while. avoids some problems with parallel communications.
all USB devices are as the name implies work with serial connections. So just about anything you plug into a computer these days works with serial ports.
most wired networks are serial (ethernet for example)
most newer hardrives and optical disc drives are SATA and these use serial communications
if you are talking about the 9 pin serial port.. hmm use to be used for various devices.. such as modems, mice, barcode scanners just about anything you can think of.. but most newer items uses the USB instead.
you will find parallel communications mostly in newer wireless devices (they can't really increase the bandwith of a single channel so that's an obvious way out).
note that the usual moniter port is neither serial or parallel.. it's a analogue connection.
parallel ports were used for more bandwidth, but nowdays serial ports are just as good or better....