What is the difference between AT89C51 and AT89S51?
How do you connect these to your computer? Which one would be easier to burn?
Answer
The information you seek can be found by comparing the data sheets: AT89C51, AT89S51, however you will see in this first link that it states:
Not recommended for new designs. Use AT89S51.
So that probably makes your last question irrelevant, you should use the AT89S51 regardless.
With regard to your second question (which I interpret as "connecting to a development host"), I suggest that you use an off-the-shelf development board unless you have the means and skill to produce your own. In-system programming can be achieved through the UART (there's an app. note in the link above), so you'll need a PC with an RS-232 serial interface and a NULL-Modem cable (the latter may come with the board). If you do not have a serial port on your PC, get a USB serial port adapter.
Obviously you need the development tools; at leas a compiler, and for debugging you'd benefit from an emulator; again the link above has references to tools.
Overall; read the manufacturer's documentation is the answer to all your questions! ;-)