You WANT a wireless adapter or your only option is a wireless adapter. All wireless adapters are subject to interference which causes random delay and packet loss. Most online games do not tolerate either of these well.
So if you can run a ethernet cable you should.
After that the adapter is only half your problem. It does not good to buy a adapter that has features not supported by your router. Also speed is not the critical component in game play. Most games use well under 1mbit/sec. Besides even the slowest wireless is faster than most people internet connection so unless you stream video INSIDE your house pretty much any speed adapter will be fine.
The key part nobody will really be able to tell you is how good the coverage will be in your house. If you read the reviews every adapter on the market is loved by some and hated by others. Mostly this is because all of the better adapters have the exact same radio chips and the exact same coverage and the difference are people houses.
Now I could like the previous post just link some device with no explanation but I don't since there really is no right answer. It is almost impossible to predict. But some people think the bigger numbers is always better that is how the vendors con people into buy things they don't really need.