It’s not usually the data rate of your Internet connection per se but the latency along the route your computer ends up tracing over the Internet to the game server, compared to the same thing for your opponent. However a very fast data rate often does result in lower latency.
You can use a program such as the command-line program “ping”, and/or in-game ping-reporting tools to see what your own latency is, and often games will show the latency of other players on the same server. Of course, ping can and does vary from moment to moment.
The latency is the delay (usually shown in milliseconds (ms)) for data to get between your computer and the server. So if your ping is 200ms, that means when you take an action in the game, the server can’t possibly know about it for 200ms (which is 1/5 of a second), while if your opponent is only getting a 500ms ping, his commands to the game will take at least ½ second for the game to receive them. Also, anything that the server determines happens in the game won’t reach you until the message can reach your computer (again, at least that ping latency time).
In practice, it’s a bit more complicated as different games have different strategies for trying to mitigate these delays. One popular technique for real-time games is to show you what would happen if there were zero latency, but then it has to correct that when it gets a contradictory update from the server. That means that in the interest of giving players immediate responses to their commands, it’s actually guessing about what will happen, and may end up correcting that a bit later, and/or showing weird things where it looks like you are doing something to an opponent accurately, but the server doesn’t think so.
So clearly, in real-time games, you really want as low a ping/latency as possible.
Oh another weird sort of thing that happens with compensation techniques some games use, is that they try to make things fair for the player with the worst connection. This can lead to a player with a slow connection messing up the experience of everyone else who has faster connections, and in these games often the experienced players will vote to kick such players off, and/or set a minimum ping requirement to join and/or stay on the server.