GEN is short for Generation. It's a long word, so it's shortened.
If your tamagotchis grows to be an adult, and it mates with another tamagotchi character, it will have a baby. When the parent leaves the baby, and you start taking care of it, it will be your next generation. If that tamagotchi lives to be an adult it can have a baby and you can start another generation.
If you're a really good caregiver you can get a lot of generations. But the generation number only goes up to 99, as with weight and age.
If you get up to the 99th generation, and that tamagotchi has a baby, it will be another generation, just not counted in your tamagotchi's status report.
GotchiGirl96