A binary variable takes one of two values, typically coded as "0" and "1".
In a broader sense "binary variable" is a synonym of "dichotomous variable": any variable that can take on only one of two values. In a narrower sense it refers to dichotomous data coded as "1" or "0". (Sometimes "1" is supposed to mean "is present" and "0" to mean "is absent", which may require handling the two values asymmetrically in some statistical analyses (see e.g. Jaccard indices).)
A binary response variable occurs as a result of Bernoulli trials, whose analysis commonly involves contingency tables or logistic/probit regression.
The term 'binary' also refers to data stored as machine-readable binary numbers rather than numbers recorded in strings of ASCII (or Unicode, or other human-readable) numerals.
In econometrics binary variables are also called dummy variables.