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It has been reported that perhaps half of nearby sun-like stars are found in multiple systems.
Why is this so common, and how do planets form in such a situation?
Kaitlin M. Kratter, “Sibling Rivalry Begins at Birth,” Nature, Vol. 518, 12 February 2015, p. 173
You are correct - when we look at stars in the sky, about half of them are actually binary systems, with a range of separations. The frequency of binarity is strongly mass dependent; the 50% figure is appropriate for stars like the Sun, but it appears to be closer to 100% for high-mass O-stars and as small as 20% for the lowest mass M-dwarf stars. There is also a change in the separation (and corresponding orbital period) distribution. The distributions of separations looks "log-normal" (like a Gaussian when you plot log separation on the x-axis). For solar type stars the peak is at around 50-100 au, but binaries become more compact (maybe 10-30 au) for low-mass stars. There is a tail of binaries with much smaller and larger separations.
These observational data are explained (or are in the process of being explained) as a combination of nature and nurture. The basic process of star formation results from the collapse and fragmentation of molecular clouds. If a collapsing fragment has significant angular momentum then the most energetically favourable way it can continue to collapse whilst conserving angular momentum is to sub-fragment into two pieces orbiting each other. This is thought to be the most basic binary formation process. Other processes include the formation of low mass companions through instabilities in cicumstellar disks or interactions between young stars in dense protostellar environments. Exactly how this leads to the mass dependence of the binary frequency and separation distribution is a topic of contemporary research; there is no definitive answer I can give.
After that I think there are two basic ways, or rather two basic locations, in which planets could form in a binary stem. Either around one or other of the stars or as circumbinary objects. The likelihood of each will depend how and when the binary formed, the binary component separation and mass ratio.
The planets would likely form in the same way that they do around single stars. If the binary star components are originally formed close together, there could be a circumbinary disk from which a stable planetary system formed. Both circumbinary disks and circumbinary planets have been observed.
Disks around individual components are truncated by tidal forces at a radius roughly corresponding to a third of the binary separation. Inside this radius it may be possible to form planets. Thus widely separated binaries could have planets orbiting the individual stars.
Alpha Centauri, which is a binary star, is likely to have planets (they may have been detected, but not everyone is convinced). The most likely scenario for planets to form stably would be if the two stars were widely separated. For instance, a pair consisting of a sun-like star and a small red dwarf at a distance of 50 AU apart would allow plenty of room around the sun-like star for planet formation. In fact, if there were a 90 Jupiter-mass red dwarf orbiting our sun at a distance of 50 AU, it would have less gravitation pull on us than Jupiter does.
