The term "identity" itself can best be seen as an umbrella under which a bewildering number of definitions and approaches can hide. So let us just sketch some major diacritics here, to which most researchers will agree.
Big versus small identities
There is a long tradition in social sciences and humanities of talking about "big" identities, such as nationality, age, gender, race, ethnicity, social class, religion, education levels and so forth. The suggestion there is that anyone can be identified (i.e. "given" an identity) by associating a person with the different diachritics. Thus, X can be described as "French, middle-aged, man, white, Breton, working class, catholic and professionally skilled". Such identity categories are collective, but individuals are seen as "socialized" into these different categories in a way that gives them an individual identity too.
Apart from that, there is a literature in which such big identities are not rejected but blended with a wide range of "small" identities. Our French middle-aged man X is not just catholic and zorking class, but also a fan of heavy metal, politically conservative, a convinced meat-eater and an amateur of fine French vintage cars. All of these "small" identities can be observed in everyday life, and they are often more important in the eyes of those holding them than the "big" ones.
Inhabited versus ascribed identities
Identity is not just what we ourselves construct, but also how others see us. Concretely, we can have s strong self-imagination of being a, b and c, but unless others also grant us these characteristics, we don't really have them.
The first aspect of identity - the things we do ourselves and the ways in which we see ourselves - is called "inhabited identity", the kind of identity in which we live. The second aspect - how others see and treat us - is called "ascribed identity", the kind of identity by means of which others "identify" us.
"Having" versus "doing" identity
There are essentialist approaches to identity, in which identity is seen as a fundamental set of qualities, often seen as stable and "determining"(i.e. they are so important that they tend to overrule all sorts of other features), and in which such essential features are seen as "held" by individuals. One has identities a, b and c.
Other approaches are action-centered and look at how we "do" identity, how we "perform" ourselves in interaction with others, how we use elaborate "scripts" and "formats" to do so, and how all of this involves massive amounts of "identity work". "Being" involves "doing". Certainly in the literature on digital culture and superdiversity, the latter action-centered approaches are important.