The symbol is always seen as Duke Nukem's belt buckle and is often seen on some of Duke's weapons. It also appears on his vehicles as well as some barrels and crates in the games.
The Nukem logo is an inverted radioactive trefoil symbol, the internationally used hazard symbol used to alert to the danger of radiation.
Duke Nukem 1 & II
The Nukem symbol is only seen a few times in the first two games as logos on radioactive barrels.
Duke Nukem 3D
Duke Nukem 3D was when the Nukem logo became the iconic symbol it is today, appearing for the first time on Duke's belt. It was used as the switch to end levels and continue onto the next level.
Duke Nukem Forever
12 years after the events of Duke Nukem 3D, Duke Nukem has become the most famous man to ever live, and the Nukem symbol has become the most iconic and popular symbol on the planet. It had been touched up and polished into a gold plated symbol for Duke Nukem Forever. Duke Nukem changed his Nuke belt buckle from a gold plated square with a branded in symbol to a solid round grey Nuke symbol.