The Pentium III is a pretty outdated CPU which will probably only run old games like Quake, the Cube engine, Doom, etc. Your graphics isn't that great either, 32MB of video RAM is way behind the times.
You can go to http://www.canyourunit.com to see a list of games you will be able to run.
A P3 machine of that speed should run plenty of games. Obviously, nothing modern.
I believe since I had to "game" with a 350Mhz K6-2 for a number of years, I can offer some relevant (if anecdotal) recommendations for games.
For the record, the machine I had was:
AMD K6-2 350Mhz 3D-Now processor
MVP3G-M (iirc) motherboard
Video Card varied; it came with a ATI 3D-Rage Pro Turbo AGP (8MB VRAM)
Sound card was some standard Creative Labs PCI card.
Network card was a SMC card, but I hardly had that machine connected to a network.
512MB of Memory, but found it faster with 256MB due to limitations of the cacheable DRAM size in the chipset.
Games I played without issues:
Duke Nukem 3D
Quake
Quake II
Quake III
Need For Speed II, III, High Stakes, Porsche Unleashed
NES emulation (only emu that ran at a reasonable speed (aside from NESticle in DOS mode) was jNES)
SNES emulation (with the Rage pro card, I usually got 18/60 fps; however after upgrading the card I got 40/60 which was a lot better)
And of course Doom (which I played using the "Doom Legacy" Windows Port) ran fine. Most DOS games would as well, but I avoided DOS versions like the plague on account of running them on a windows system being a gigantic pain in the *censored*. (Also, I had a laptop that ran DOS/Win31 that was perfectly setup for running DOS games via a CONFIG.SYS menu choice, and a 75Mhz Pentium was plenty for most DOS games).
Now to be fair the framerates probably weren't what a lot of people would like. Quake III was pretty much unplayable even to me with the Rage Pro.
However, After I upgraded to a Radeon 7000, I could play nearly any game I had installed at full speed. Naturally, depending on how the games were written, some of them bottlenecked on the CPU, but a lot of the processing for graphics took place on the Graphics card which meant games became a lot more playable. (And, most importantly, it actually supported OpenGL properly, unlike the ATI card whose OpenGL Support was... well, hilariously inadequate).
Given the information you've provided though, it's difficult to really compare. If the graphics is on-board, it might be using "host-based" processing, same with the Audio, which basically means it uses the CPU as a DSP and GPU, which takes processor capability away from everything else. At 1Ghz, though, you ought to have some to spare, particularly with older games like those I've mentioned. the 32MB Graphics card is hardly a limitation in that context, since 32MB was considered only something a "high-end" gamer would have in their machine around the time the system was released. And, since my own machine ran with a meagre 96MB of RAM (under windows XP) and I didn't have huge issues with the games above, memory is probably no issue either.
Aside from the games I've mentioned above I can't really offer many recommendations. What you can do is look up some games released around the 1998-2001 era, which will work perfectly on that machine, and see what you like.
EDIT:
You can go to http://www.canyourunit.com to see a list of games you will be able to run.
On another note, I avoid this website like the plague and refuse to recommend it to anybody. For one thing, it says that I can't run half the games I have installed (Just Cause 2, GTA IV) but they run perfectly fine. And it says I can run other games that have higher requirements (Saints Row 2) just fine. Another point is that the site caters more to modern games. Note that the only Quake Game in their list is "Quake 4" and the earliest Need for Speed game in the list seems to be "Most Wanted", this paired with the sites complete inability to actually give accurate output makes me reluctant to do anything but dismiss it as a useless sideshow, at least in it's current form.
EDIT 2:
Another interesting tidbit with regards to the K6-2; while typically it would run fine, I actually managed to get it to overheat on at least 3 occasions. (the little beepy siren went off) after rebooting I would find the CPU temp often in excess of 90 degrees. Turns out (after replacing the CPU years later) that the little thermal pad between the CPU and heatsink and practically rotted away. Strangely it only ever overheated on a single level of a single game, the "Overlord" level of Duke Nukem 3d. Even stranger, it was always after I opened a specific door. Odd.
