You have had the warnings and furthermore those P3's ran really hot (it was part of the reason for the P4 coming out relatively quickly) and as mentioned overclocking can drop the life of a CPU somewhat (if you keep it cool though then it should be OK, search for electromigration as that is the most common cause of failure and not helped at all by heat).
There are two methods
Front side bus clockspeed and multiplier changing. Intel general does not allow multipliers to be changed on anything but the most expensive of their range which leaves you with FSB.
Secondly if this is a branded (dell, HP, time, acer.....) machine they usually sell the machine with a seriously stripped down BIOS that prevents overclocking (one of the many reasons I build machines for people).
Occasionally they also allow BIOS upgrades to allow tweaking and "hidden" BIOS tweaking screens but I would not count on it.
GUI programs are not going to happen for such an old machine.
Still the frontside bus (henceforth FSB). This ultimately controls the ram and the CPU speed (modern machines allow an unlinked mode but a increase in older machines affects them both hence people suggesting good ram for overclocking (I suggest good ram for all machines but that is besides the point).
Increasing this increases the clockspeed.
Voltages. Transistors work by creating an electric field and attracting or repelling charge carriers depending on transistor type which takes time (the distance across stuff that has to be changed is what gives you the 90nm,130nm, 65nm.... measurements you might see with chips), higher speeds mean there may not be enough to time to get the carriers there and the chip will crash (just one hiccup can take things down). A higher voltage means a larger field and shorter response time.
A higher voltage also means a higher temperature (see first paragraph) which means it needs more cooling. Trying to find more/better than stock cooling for a machine as old as that these days (assuming you do not want to mod/make stuff and frankly if you have to ask this question making cooling is probably not something I would suggest right away) will probably be about as expensive as a decent second hand machine.
Ram timings. Assuming you can not run in unlinked mode the ram will have a set of timings associated with it (probably 2.5, 2, 3,10 or something similar). These refer to various things but faster speeds mean you up the voltage (with above problems) or lower the timings (making for a slower machine owing to the now slower ram). Again vendors do not like people doing this and usually block it.
As already mentioned those older machines do not overclock well (especially the P3, AMD was the king of overclocking during that time and up to the core2 line really). This together means your 800MHz machine will probably be lucky to hit 880MHz (more than 10% is practically unheard of outside of the liquid nitrogen/helium circles).
