In general, it can be pretty unpleasant to have to fire someone. Most of the time, you don't even want to do it, but whether it's due to expense or repeated workplace mishaps, you feel you have no choice. An article in Forbes walks potential terminators through doing it with minimal awkwardness and hurt feelings. 

While the article goes into things in more detail, in short you want to make sure that, when you do fire someone, it's not a surprise: they should have had many previous opportunities to improve. When the time comes you should be prepared and avoid going off-track: know what you need to say and say it. Remember, too, to be respectful. Also, if you can, try to provide what the article calls a soft landing, that is some sort of package that helps them through the transition. Finally, if possible, avoid firing someone at all: offer them the chance to resign instead.