Rather than picking one, I’d like to mention three.
The first is Fomalhaut b, it’s pretty much the first time we’ve been able to actually picture a planet around a neighbouring star (http://arxiv.org/pdf/1210.6745v2.pdf, or http://en.wikipedia.org/wiki/Fomalhaut_b). The thought that the light in this photograph actually came from such a faraway planet is, well, staggering.
Next is Epsilon Aurigae. Every 27 years it gets darker. It looks like it is surrounded by a debris disk, but with a gap in the middle, but we’re not sure and whichever way we look at it, it’s pretty unusual.
And the last one is Betelgeuse, one of the bright corner stars in Orion. This star is massive and highly instable and at some point in the near future (well, sometime within the next thousand years or so) it should explode in a supernova. When it does it will be visible in daylight for a couple of weeks and will cast a shadow at night. I kind of hope I get to see this.
My favorite star is Proxima Centauri, it is our closest star (other than the Sun), but currently we have no way of reaching it (we would have to travel at the speed of light for 4 years to reach it)… My favorite planet is saturn- it is amazingly beautiful, and also has a moon Europa (correct me if I’m wrong…), that could harbor life between its hot core and huge ice sheets covering its surface.