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 stars.
Binary (or more) star systems are said to be common: the best number I've found is that roughly third of all star systems have two or more stars. They are extremely 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.
A system can have three or more stars. The term Multiple Star System is sometimes reserved for this case.
A common classification of binary stars is based on the method by which they were determined to be binary:
Another classification is based upon how close they are and how much they interact:
A binary star's Mass Ratio (μ) is the ratio of the two masses, i.e., 1 for stars of equal mass. When the Total Mass can also be determined, e.g., from the orbital period and size, the mass of each star is evident.
Binary stars generally have similar composition (as shown by their spectra), as if they were formed together.
Given the large scale of distances between the stars, the different sizes/Spectral Classes of the individual stars binary stars show a wide variety, and the interactions produce characteristics unseen in non-binary stars. For example:
Black Widow Pulsar (B1957+20)
Black Hole Binary (BHB)
Double-Line Spectroscopic Binary
Hulse-Taylor Binary (PSR B1913+16)
Stellar Population Synthesis (SPS)
Stellar Distance Determination
Stellar Parameter Determination
Transit Timing Variations (TTV)
Turn-off Point (TO)