The concept (as with so many philosophical concepts) goes back to the medieval period. It is clearly articulated by Henry of Ghent (Quodlib. 5. quest. 8) as being a 'double negation', and by Scotus (Ordinatio II. iii. q2) as 'privatio divisionis in se et privatio identitatis ad aliud' - the privation of division, and the privation of identity with another.
Taking the first of these negations (I will discuss the second elsewhere), what does 'a privation of division' mean? It is where we get the word 'individual' - literally, the undivided or undividable. What is meant is that the division of genus into species, the division of the species into sub-species, and so on, comes to a halt when we reach the individual. We can divide a genus (e.g. animal) into different species of the same genus, such as giraffe, koala, swan, man, and so on. A species such as man can be divided into individuals of the same type, such as Socrates, Plato, Aristotle and so on. But there it ends. You cannot divide Plato into different things of the type Plato.
This is all rather medieval, but it has a simple logical consequence. It means that the inference
A man is white and a man argues, therefore something that argues is white
is invalid, for Socrates might be the only person who is white, and Plato the only person who is arguing, and so no one individual is both arguing and white. The inference fails because there can be always be different things of the same kind, for anything above the 'most specific species' (i.e. the individual). But on the other hand
Socrates is white and Socrates argues, therefore something that argues is white
is valid, for there cannot be one Socrates who is white, and another different Socrates who is arguing (at least when 'Socrates' is understood in the same sense). A name, taken in the same sense, cannot be verified of different individuals. Or rather, what we mean by 'individual term' is such that inferences of the type above are valid. If 'A is B and A is C' implies that some B is C, i.e. if it cannot be true that A is B and A is C, without it also being true that some B is C, then 'A' must be a singular or individual term. The semantics of the singular term is defined - I will argue wholly defined - by this feature.