In commodity trading the expression "basis risk" typically refers to differences between the commodity you trade in the spot market and the commodity on which futures are priced. This difference causes hedging to be imperfect.
The example usually given is a farmer who produces wheat in Kansas but hedges with Wheat Futures traded on the Chicago Mercantile Exchange. The prices of these are NOT going to be exactly the same, even at maturity. The reason is that there may be differences in the grade of wheat that the farmer is growing compared to the kind of wheat that is quoted on the CME. Furthermore the CME wheat is delivered in Chicago, so there would be a transportation cost to send the wheat from Kansas to Chicago resulting in a small price difference. So $S_T \approx F_T$ instead of exactly equal.
Another example is the hedging of fuel costs by an airline. I am not a pilot but I understand that you cannot take Heating Oil received at delivery of HO futures and put it directly in the tank of an aeroplane, airplane fuel is a very similar product but not exactly the same. So the prices of airplane fuel and heating oil are very close but not identical. And the difference is not constant but should be modeled as a random process.
Another example: you have 9 year Treasury Bonds but hedge them with ZN, the 10 year bond futures, etc. Or hedging AAPL stock with Nasdaq futures, etc. etc.
In summary then "basis risk" is a risk which causes hedging with futures to be imperfect in a given situation.