Unlike The Sun, quite a few stars are mutiple star systems, that is in addition to the main star, there is at least one orbiting star. The centre of rotation may be the bigger star or a spot in between the two stars which they rotate round.
Nearly everyone can remember when Luke Skywalker at the end of a hard day looks up to the sky and sees two stars setting. Tatooine, the planet he was on in Star Wars IV - A New Hope would have been operating in a binary star system. An Extrasolar Planets (Exoplanets) can exist in a multiple star system. The star Algieba in the constellation of Leo is both a mutliple star system and one that has exoplanets.
Binary star system is another way of star a double star system and just refers to two stars that orbit one another. The abstract term for two and more stars is a Multiple Star System. The most number of stars in a multiple star system is 7 and that title is jointly shared by AR Cassiopeiae and Jabbah. It seems slightly apt that Jabbah has a bloated star system after all the similarly sounding Jabba, the Hutt from Star Wars was bloated.
Two stars could look to be double stars but they`re not because they don`t rotate round a fixed point in space and they`re millions or billions of miles apart to exhert any influence on one another.
The word Binary refers to two so its logical that binary star system refers to two stars orbiting either a fixed point between the two stars or one orbiting the other. Any more than two is a triple star system (consisting of three stars) or multiple (anything greater).
The most well know star that is a Binary Star System is Sirius. A Binary Star System is also known as a Double Star System. The Sirius star system consists of a white main sequence star and a blue dwarf star. In the case of Sirius, the stars are one large main and a smaller child. However, the star system can be of the same size and be of equal power so that they both orbit a central point whereas in the case of Sirius, Sirius B orbits round its larger neighbour.
Our Sun might have been a mutliple star system but Jupiter did not gain enough mass to start nuclear fusion to become a star. It is also said that the elements that make up the planets, the carbon, etc can only have been created from resulting debris of an exploding star going Supernova so if the Sun had been a double star system then the companion star has all gone now.
There has been a theory that our Sun is orbited by a brown dwarf that has been called Nemesis which is responsible for a number of Mass Extinctions in the past. As it is a brown dwarf, its not easy to spot. Nemesis when it gets closer to the Sun, it causes Comets that reside in the Oort Cloud to begin to make their way in wards.
Equal Mass binary system has two stars of equal mass hence the name orbiting a spot in between the two locations. One of the stars would not orbit the other like the Earth orbits the Sun. An example of an equal mass multiple star system is Rigil Kentaurus better known as Alpha Centauri in Centaurus.
A star which orbits round another star like Earth orbits round the Sun would be an Unequal Mass binary system. The centre of an Unequal Multiple Star system would not be the centre of the larger stronger alpha star but off centre because the smaller would have enough influence to pull the the larger star off centre slightly. When Earth and Jupiter orbit the Sun, they are able to pull at the Sun and pull it off centre a little. An example of an Unequal Multiple Star System is Foramen in Carina.
The third type which will consider for now is the Double Binary system which will have two sets of binary stars orbiting a centre in between them. In the outer binary system, they will orbit round a centre and then those two will orbit round a fix centre with another two. Mizar is a Double Binary star system which has four stars separated into two pairs and they interact with Alcor.