Have you ever played a game where someone tells you to count to 10, but they're going to stop you halfway through and say "halfway"? That's kind of like what "half a life" means in Star Trek: The Next Generation.
In one episode, there's a planet where people are older than they seem because they use a special machine that makes them live for a really long time. But there's a catch - when they turn a certain age, they have to go through a process called "Resolution" where they basically die.
This is where "half a life" comes in. One of the characters, Lwaxana Troi's friend named Timicin, is about to go through Resolution. But instead of accepting this, he asks to go on a special mission to try to save his planet from a disaster first. While he's on this mission, he meets a woman named Lwaxana who he becomes very close with.
But when they return to the planet, they find out that Timicin still has to go through Resolution, and he doesn't want to because he's fallen in love with Lwaxana. So they try to find a way to save him instead.
Essentially, "half a life" means that Timicin has already lived for a long time, but he's still only halfway through his expected lifespan. And now he has to decide whether to accept his fate and go through Resolution or try to defy it and stay alive.