The simplest and most concise breakdown of modernity and Modernism I have come upon, this from Timothy R. Quigley of the New School

From Modernism to Postmodernism: Three Key Concepts

1. Modernism is generally used as a way of referring to an aesthetic approach dominant in European and American art and literature in the Twentieth Century. The principles of formalism and the autonomy of art are key features of modernism.

2. The "project of Modernity" can be thought of as the development of science, philosophy and art, each according to its own inner logic. [See Habermas, "Modernity - An Incomplete Project"; cf. Greenberg, "Modernist Painting".] This links the concept of modernity to the concept of modernism as it was articulated by Greenberg.

3. The concept of the avant-garde is that of a loosely organized oppositional force and challenge to the dominant artistic culture. The avant-garde is often thought of as part of the "inner logic of modernism" - the built in source of contradiction or critique that moves art forward. (Note that this assumes a model of progress as part of the inner development of the arts and culture.)

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