Modality is a way of formally, logically talking about what I would call the higher-order aspects of the ways of being of things. It is most commonly associated with necessity and possibility, but I think these are actually atypical examples that may give a misleading impression of what modality in general is, because necessity and possibility both have a kind of extreme all-or-nothing character that does not hold for modality in general.
I don’t believe quantification across all possible worlds could be interpretable by any process of interpretation, so I don’t consider it even intelligible in an acceptably strong sense, and I also don’t believe in unconditional necessity. Anything real or any truth about it, as well as anything I would accept as a legitimate formal construction, has conditions, even if they are only implicit. So, standard modal logic concerned with operators for these unconditional things — technical interest aside — does not seem very useful to me, because the resulting propositions would be too strong.
I am a bit surprised that Brandom is so charitable toward formal possible-worlds semantics, given his reservations about formalism in general. Technically innovative as it was, this approach seems like an extravagant extreme of infinitary classical representationalism.
Modality itself should be safe from these concerns. At a handwaving level, I imagine an indefinite number of modalities related to particular specifiable conditions, and expressing structural “degrees” or “flavors” of more specific necessity or possibility based on those conditions.
Wilfrid Sellars suggested that modalities should be understood as specific forms of normative bindingness. This seems very helpful as an alternative to extensionalist possible-worlds formalisms. (See also Why Modality?; Modality and Variation; New Approaches to Modality; Deontic Modality; Redding on Morals and Modality.)