First a history lesson (if that's OK):
The serial port was originally designed to connect a computer to a modem (back in the days before computers had internal modems). The original standard (V24 in Europe & RS232 in the USA) used a 25 way DIN connector. These 25 pins, however, included a lot of old fashioned signals that are not used by a modern modem, so computer designers started to use a 9 pin DIN connector for serial ports instead and you won't find many 25 way serial ports around.
Because the serial connector is designed to connect a modem to a computer, the transmit output signal on the computer is an input signal into the modem (and vice versa). Hence, if you connect all the 9 pins of a serial port on a computer dirctly to all the 9 pins of another computer, it won't work, because outputs will connect to outputs etc. Thus, to connect two computers via their serial ports, you need a ''null modem'' cable (''null'' because there's no actual modem there) which crosses over the transmit and receive pins, etc.
The parallel port was originally designed to connect a computer to a printer. Although the connector standard for this was a rather large and ungainly ''Centronics'' connector (you'll still find them on the back of parallel printers), computer manufacturers decided to use the same 25 way DIN connector that was originally used for serial ports. Confusing!
USB is a much more modern standard for connecting anything from printers to cameras to a computer. It's a very fast serial connection. USB comes in three flavours, by the way, (1.0, 1.1 and 2.0), but that's not too bad as the later ones are compatible with the earlier ones. At the computer end, the USB connector on the cable is a fairly small rectangular plug and at the peripheral end of the cable it's either a square plug with two corners cut off or a really small miniature plug. For the same reason that you can't connect two serial ports together, you can't connect two USB computer ports directly together or they would fight.
Most recent laptops don't support either serial or parallel ports, but all support USB. Most desktops still support parallel and serial ports as well as USB(although I guess it'll get rarer).
The upshot of all the above is:
On the computer, a 9 way connector will be a serial port and a 25 way connector will be a parallel port It would have to be an old modem or a **really old** computer to have a 25 way serial connector.
To connect two computer 9 way serial ports, you must use a ''null modem'' (i.e. crossed over) cable. The easiest thing is to tell the assistant at the computer store what you want it for and he/she should get you the right type. Similar cross over cables with a 25 way connector at each end can be obtained for connecting two parallel ports, but they're not very common. Beware of a cable with a 9 way connector at one end and a 25 way connector at the other - it's almost certainly to connect a computer with a 9 way serial port to an old modem with a 25 way serial port. It won't convert serial to parallel!!
Because laptops often don't have either serial or parallel ports, you can buy an adaptor which converts a USB port to a serial port or one which converts USB to a parallel port.
You don't mention what the other computer is. Does it have a serial port? If so, the solution might be buy a USB to serial convertor to get from USB to serial on the computer which doesn't have a serial port and a null modem cable to connect the convertor to the other computer. If neither computer has a serial port, you could use two USB to serial convertors (one on each computer) and a null modem between them, but that's really getting a bit messy.
Shopping list to connect two computers via serial ports would be:
a) Both computers have a serial port: buy 1 null modem cable.
b) Only one computer has a serial port: buy 1 USB/Serial convertor and 1 null modem cable.
c) Neither computer has a serial port: buy two USB/Serial convertors and 1 null modem cable.
It's a bit of a dilemma: connecting the two computers together, if you can, will give you an easy software life (especially if both are running XP). Getting the computers networked (via radio or ethernet) takes a bit of setting up but can pay for itself in sharing printers, files and network connections. Using something like a USB memory key is a very simple way to transfer files from computer to computer but means you have to find all the files you need and probably do some of the transfer of settings and preferences by hand.