A binary star is a pair of stars that orbit each other.
Double star means the same thing except that it also includes
stars that are not orbiting and not close together but happen to
be on the same line of sight from Earth, referred to
as apparent binary or optical double stars.
A star's binary companion is a star with which it shares such an orbit.
A higher-order multiple star system has three or more stars
and the terms multiple star system and multiple star are sometimes
used with that meaning, but often meant to include binary stars as
well.
Binary (or more) star systems are said to be common: the most
convincingly cited number I've found is that roughly third of all
star systems have two or more stars. They are extremely useful in
the study of stellar physics, both to use the orbital dynamics for
stellar parameter determination, and for those close enough to interact
further, giving additional situations to observe, to infer and test
the physics of stellar structure. Their radial velocities'
are sometimes both determinable, yielding their mass ratio
(μ, the ratio of the two masses, i.e., 1 for stars of equal
mass). In cases where the total mass can also be determined, e.g.,
from the orbital period and radius, the mass of each star is evident.
The commonly-used system for designating the individual stars of a
binary/multiple star system consists of adding letters to the star
system's designator, with letter "A" for the brightest, "B" for the
second brightest, then "C" and so on. For example, the two
stars making up Sirius are termed "Sirius A" and "Sirius B".
If two are very close and a third is distant, the two close stars
might use lower-case suffixes, i.e., "Aa" and "Ab", with the further
member called "B".
Binary stars generally have similar composition (as shown by their
spectra), as if they were formed together. Binaries formed together
are known as primordial binaries, another sign being aligned
rotation axes. A capture requires the coincidence of stars passing
close to each other, plus something to change their velocity, such
as a third star, or with just two, tidal forces between them
(i.e., tidal-capture binaries). The "close pass" is more
likely in areas with a very high density of stars such as the center
of globular clusters or galaxies. Simulations have suggested
that tidal capture binaries are often the result of an exchange
between a single star and an existing binary resulting in the
single star becoming a binary companion, with the replaced binary
companion becoming a single star.
A common classification of binary stars is based on the method by which
they were determined to be binary:
visual binary - separate stars can be seen or photographed.
eclipsing binary - a periodic variation in luminosity suggesting a star passing in front of its companion.
spectrum binary - a star whose spectrum looks like the combined spectrum of two stars.
astrometric binary - measurement of the star's position in the sky over time suggests an orbit-like movement around an unseen companion (i.e., not bright enough to be seen).
spectroscopic binary - measurement of the spectrum over time shows an oscillating Doppler shift of some spectral line, indicating the type of radial motion that orbiting a companion would produce.
Another set of classes is based upon how close
they are and how much they interact:
compact binary - small orbits: one criteria is that they orbit in five days or less.
close binary star - similar meaning: close enough to consider their cross-influence and interaction.
detached binary - sufficiently distant that the stars are (basically) spherical.
semidetached binary - sufficiently close that one of the stars transfers mass to the other, but not touching.
contact binary - touching; at minimum, both stars "stretched" by gravity to a contact point.
Given these possibilities as well as the different sizes/spectral types
of the individual stars, binary stars show a wide variety and
interactions between them produce characteristics unseen in non-binary
stars. For example:
VV Cephei systems - (like VV Cephei) consisting of a giant star and a hot star forming a long-period variableeclipsing binary with an eccentric orbit and which interact when they are near each other.
ζ Aurigae systems - (or Zeta Aurigae systems, like the star ζ Aurigae) are analogous to VV Cephei systems but with a different class of giant.