4.25 light-years away
Proxima Centauri
What is the closest star to us, and could anything live there?
No star is closer, and none is likely to be for the next thirty thousand years. It is also invisible to the unaided eye, which tells you something about how little brightness has to do with distance.
A star nobody saw until 1915
Proxima is a red dwarf about an eighth the mass of the Sun, radiating so feebly in visible light that it went unnoticed until Robert Innes picked it out from Johannesburg in 1915. At magnitude 11 you need a telescope. The two bright stars beside it in the sky, Alpha Centauri A and B, had been famous for centuries; the nearest star of all was hiding in their glare.
It sits about a fifth of a light-year closer to us than they do, and orbits the pair on a circuit of roughly half a million years. Whether it is truly bound to them or merely passing was argued over for decades; the current answer is that it is bound.
A temperate planet, and a difficulty
In 2016 a planet was confirmed around it: Proxima b, a little more massive than Earth, orbiting in the zone where liquid water could exist on the surface. That zone is very close in — the star is so dim that the planet completes an orbit in eleven days.
The difficulty is that Proxima is a flare star. It brightens abruptly and violently, occasionally by orders of magnitude in the ultraviolet, and a planet that close would take the full force of it. Whether an atmosphere could survive that is genuinely open. It is the nearest place worth asking the question about, and the honest answer is that nobody knows.