If you are building a general purpose or a fighting gun, you want a barrel with a 5.56 NATO chamber. If you are building a match grade rifle, you may want to consider a barrel with a .223 Remington chamber.
.223 Remington chambers are tighter than 5.56 NATO chambers, and 5.56 ammunition will not reliably feed in a .223 chamber. .223 Remington ammunition will feed without issue in a 5.56 NATO chamber.
You may also come across hybrid chambers like .223 Wylde. The .223 Wylde chamber was designed as a match chambering for semi-automatic rifles, and will accomodate both .223 Rem and 5.56mm NATO ammunition. It is relieved in the case body to aid in extraction and features a shorter throat for improved accuracy.
I don't make jokes, I just watch the government and report the facts. -Will Rogers
An appeaser is one who feeds a crocodile, hoping it will eat him last. -Winston Churchill