If your computer is too slow or does not offer high enough resolution, your are at a disadvantage. I can't tell you what is good enough, but I use a 133 Mhz pentium with 32 Megabytes of memory and a 17 inch screen. I also play on a laptop with a 166 Mhz pentium, 32 Megabytes of RAM and a nice active matrix display. In both cases I set the resolution (with AoE) to 1024x760. (I found that my play improved considerably when I noticed this option.) I also set the scroll rate fairly high.
Your connection to the internet affects the lag (which affects both you and the other players) and hence affects your (and their) enjoyment of the game. But, it does not seem to affect your ability as compared to them. In other words, if your connection is bad, both of you suffer similarly. Right now, a 28.8 modem and a good internet provider is the standard. But, within the next two years isdn or cable modems will become the standard and those stuck with 28.8 modems will find themselves excluded from some games.