The term "habitable star" refers to those stars which, by our best estimates, are capable of hosting habitable planets. Habitable planets are defined, in this context, as those which have life, no matter how primitive.
Because we have never observed any life other than that on Earth, all our current searches consider first the zone around a star in which a planet could have liquid water on its surface (termed the habitable zone), and then consider whether the star would last long enough for life to naturally evolve there, given a timeline of evolution similar to life on Earth. Without having examples of extraterrestrial life to look at, we must assume that life on Earth evolved neither anomalously quickly nor anomalously slowly relative to life elsewhere. This constrains the range of habitable stars to those rather like our Sun, for a number of reasons.
First, habitable planets are likely to exist only around main sequence or possibly subgiant stars (those expanding due to depletion of hydrogen in their cores - stars can remain stable in this form for a long time, though not as long as their main sequence lifetime). Giant stars are actually the last stage in a star's life, shortly before it explodes or peters out. Their increased size and luminosity will fry any planets that were in its habitable zone, and the star won't last long enough in its giant phase for life to evolve on any worlds in the newly-widened habitable zone.
The very hottest stars, those of spectral class O, evolve off the main sequence very rapidly, from a few million to as little as a few hundred thousand years. This is probably too rapid for planets to even form, much less acquire ecosystems. Additionally, class O stars emit so much radiation that the radiation pressure serves to blast the material that could coalesce into planets clear out of the system. The next cooler class, B, may have similar problems with radiation pressure. Even if they don't, class B stars are still very short-lived, burning through their hydrogen fuel in tens of millions of years.
This neatly excludes the most spectacular stars, the giants and O or B class main sequence stars that form the majority of the naked eye visible stars. The next cooler class of stars, class A, can definitely host planets. In particular, the star Fomalhaut has a large, Jupiter-like world orbiting it. We haven't spotted any Earth-like planets around class A stars, but then, we haven't spotted such planets around any stars at all, and the technology that would let us is only just now coming on line. The issue with class A stars, though, is longevity. They might last one to two billion years on the main sequence, which is probably not enough time for life to evolve. Even if it is, the first forms of life will just be beginning to get a foothold when the star expands into a giant and burns the hapless planet - and all its inhabitants - to irradiated ash. So, no aliens from Sirius, it seems.
That leaves spectral classes F, G, K and M. These aren't terribly spectacular-looking in our night sky, except for Alpha Centauri, and that only because it's so close. Class F is marginal - the hottest F stars, termed F0V through about F5V, have much the same longevity problem as A stars. Cooler Fs, though - F6V through F9V - last between seven and eleven billion years, more than enough time for life, or even civilization to become established.
Class G, the spectral class of our Sun, is pretty much universally suitable. Even hot G stars like the Sun or Beta Canum Venaticorum last more than long enough. Cooler G stars like Tau Ceti and 82 Eridani are some of the strongest candidates for extrasolar habitable world hosts.
K stars have no particular problems with longevity. All of them last a bloody long time, pretty much universally longer than the universe has existed (13.5 billion years). The problem is that the habitable zone around cooler K stars might be too small. This has two implications: one, there's less space and so the probability that a planet exists in the zone is smaller, and two, the orbit might be so small that the planet would be tide-locked. Tide locking can play hell with conditions on the planet, and might preclude habitability. Still, hotter K stars, down to about K3V or maybe even as low as K6V, are good candidates for hosting habitable planets.
That leaves red dwarfs, which are class M and bottom-range class K main sequence stars. These have the small habitable zone problem described above. Also, lower-mass stars tend to flare quite energetically. Indeed, the smaller the star, the more powerful the flare is relative to the star's normal output. A flare from a red dwarf like Gliese 581 is as energetic as a flare from the Sun, which is many times more powerful under normal circumstances. These flares can bombard any nearby planets with heat and radiation, with devastating effects to any life on those planets. Ultraviolet and X-ray radiation can directly damage living cells. Particle radiation is even worse, if not deflected by a powerful magnetic field, like Earth's. Even if it is deflected, it can ionize the upper layers of the atmosphere and carry them away into space. Over time, this can strip a planet of its atmosphere entirely, which is likely what happened to Mercury, and is happening to Mars right now. (Mars has a magnetic field, albeit a weak one, and is relatively far from the Sun - at the extreme edge of the habitable zone - which slows down air loss significantly.)
All that said, there are many red dwarfs in the galaxy - several times more than even K stars, the next most common type. Some of these don't flare, or at least, flare less violently. All this means that there might be habitable planets around red dwarfs, and even if they're proportionally much rarer than habitable worlds around F, G and K stars, there still might be more of them in absolute numbers.
Stellar habitability by spectral type
- O - Hell no
- B - No
- A - Unlikely in the extreme
- F - Unlikely to probable, depending on temperature and mass
- G - Likely
- K - Likely to possible
- M - Unlikely but not impossible