A double star is simply two stars very close together (and a triple star is three stars close together). A double star can either be a binary star where the two stars are linked and move round their common centre of gravity or an optical double where they are simply in the same line of sight by accident.
Generally the two stars are close enough that they appear as one star to the naked eye. There are few exceptions such as Alcor and Mizar which are fairly easily seen as two stars as they are nearly 12 arc minutes apart. The separation between the two stars in a double star is measured in terms of angular resolution in arc degrees. Your little finger at arm’s length is roughly one arc degree. The separation of double stars is measured in arc minutes (1/60th of an arc degree) and arc seconds (1/60th of an arc minute).
The ability of a telescope (or a pair of binoculars) to see a double star as two stars (this is called resolving the star) is a function of the diameter of the telescope (or the single binocular) not magnification. The theoretical resolution of a 100 cm (four inch) telescope is 1.1 arc seconds, but in practice it is more likely to be about 1.3 arc seconds. Resolution also depends on the difference in magnitude between the two stars: the closer they are in terms of magnitude the easier it will be to see the two stars in your telescope. In particular a faint star close to a very bright star is easily lost in the glare of the main star.
Another important measure in observing double stars is the position angle. Briefly this is the counter clockwise angle between the secondary star and celestial north (clockwise if you are using a diagonal), but generally this is not useful unless you are looking at a triple and you want to know the angle between the two secondary stars.
Historically double stars were considered to be rare and thus highly prized as the calculation of the orbit of a binary star can reveal the masses of the two stars. We now know that a majority of stars are binary (triple, quadruple etc.) and the next release of data from Gaia is likely to contain the information for 60 million [sic] binary stars. Yet Gaia can only capture 60% of the 10 million binary stars less than 250 parsecs (815 light years) from the Sun.