It sounds to me like there is a pretty good chance your graphics card is malfunctioning. It may be overheating, or some other issue (like old age - electronic components do slowly wear out and I'm guessing this is over 5 years old).
At the very least, you should try a complete power down for several seconds before booting to make sure everything is actually reset. Picking a "reboot" directly from inside Windows may not be quite good enough to get everything completely reset.
It may also be a good idea to open up the case and, after restarting, make sure the fan on the graphics card is working. A bit over a year ago I had the fan on my graphics card fail and the card worked fine for desktop use but when running the game it would eventually fail in various ways due to overheating, sometimes failing completely which caused it to not put out a signal to the monitor and therefore the monitor went black and then indicated that no video input was detected (which sounds like what happened to you). The graphics card may use passive cooling (so it hasn't got a fan), in which case you need to make sure it is clean and not clogged up with dust which makes a pretty good insulator and can cause high temperatures in the thing if there is enough of it. Also check the other fans to make sure they are working, since a failed fan on the case will cause poor airflow resulting in heat built up. You can download a graphics card monitor to tell you how hot it is getting, for example GPU-Z from TechPowerUP will tell you everything about the card and can monitor all of its sensors including temperature. Different cards will run at different temperatures normally, so something like 80 or even 90 degrees C might be normal (I doubt it should go over 90 unless perhaps it is the fanless version which are typically designed to run hotter) but if it goes over 100 degrees C it is almost certainly overheating (on the other hand, the one I had fail could actually still work while reporting over 120 C for a while before it would actually fail, and I saw it go over 130 on a couple of occasions, so it might keep working just fine when only a little over 100 C).
If it is a notebook computer this is all more complicated to check. A monitor program like GPU-Z should still tell you how hot it is running.
Also, before panicking about possibly having a near-dead graphics card, run "dxdiag". This is the DirectX diagnostic tool. It is a good first check to make sure that is still working, or if it can spot an issue. There a some small chance that the DirectX installation has gotten messed up. If you are running Windows XP this tool even has tests you can run (on versions of Windows after that it only shows information, no tests are available).
It is also possible that your Civ4 installation is messed up.
So a question is can you play any other 3-D games? If they all have the same problem then it is either DirectX itself or the video card.
The good news about the video card is that if you do need to replace it then the lowest end current generation GeForce card, the GT 610, is (theoretically, anyway) probably a bit over 3.5 times as fast as your current card, has 1GB memory (2GB in some cases), and costs under $50 including shipping here in the US (I just checked Newegg and they are mostly in the low to mid 40s + $3.99 shipping) or you can go up by around $10 for a GT 620 (twice as many cores, each a bit slower, but same memory and memory bus width) or up to somewhere around $70 for a GT 630 (same core count as 620 but each slightly faster, same as the 610 I think, and with a memory bus that is twice as wide - although there is a completely different design also called the GT 630 which has many more cores of a different design, due to being a cut down newer GPU instead of the older GPU the "regular" one uses, but a narrower memory bus with slightly faster memory; I have no idea why they do stupid things like two completely different designs with the same model number). Any of these can handle Caveman2Cosmos better than what you currently have. And, of course, you can spend more to get cards that are faster, or even several times faster, but it is probably a bit pointless on a motherboard that is the same vintage as a GeForce 7300GT.
