Incoming links: Proxima Centauri
Some stars are so close that we can actually see their angles move with time due to the relative motion between them and the Sun, e.g. Proxima Centauri!
Distance of stars nearest to the Sun as function of time
. Source. 
Some notable ones:
- Proxima Centauri, the nearest one, at 4 ly. It is part of the Alpha Centauri star system, which contains two other stars at very similar distances as well, and their relative distances to earth will change positions in a few tens of thousands of years.
- Sirius, the brightest star in the sky at 9 ly