The sort of hardware you might consider depends on the purpose of the
computer interfacing.
For a very prototyping and data aquisition a good bet is some sort of
Data Aquisition card. This puts most of the problems in the software
domain and allows simple monitoring of analgue and digital inputs.
For science based projects DAQs are cheeper and less bother than
building something yourself. Moreover, they let you do most of the
work in software. A good site is
http://www.geocities.com/data_acq/vitalink.htm
If you want to do low speed interfacing to a microcontroller based
circuit you should consider RS232. Many microcontrollers can be
programmed easily down the RS232 with minimal external components.
Moreover, using a microcontroller based circuit will let you do a lot
of the work again in software. In addittion to programming most
microcontrollers will have a UART wich can interface easily with the
RS232. To get started I would recomend a commercial development kit
such as the STK500. The STK500 is a ready built hardware that can be
programmed down the RS232 and then easily interface with the PC. You
can then learn about hardware from it. The STK500 is based uses the
atmel AVR microcontroller which is perfect for basic interfacing
projects. The web is also awash with easy to build AVR programmers.
Once you have the programmer the microcontroller is easily programmed
in asembly, basic or C.
Check out
http://www.atmel.com/dyn/products/tools_card.asp?tool_id=2735
http://www.ingdm.se/avrdev/avrdev.htm
