It is useful to begin analyzing a piece of music by creating an overview of what you know and understand about how the piece is organized. The overview will help you to focus on the aspects of the piece that are most interesting and instructive, as well as creating a list of the things that you do not yet know or understand about the piece.
Music is sound whose timing, pitch, and timbre have been organized by people for social purposes other than simple communication. Understanding a piece of music, then, focuses on understanding how and why the sounds have been organized. The analysis suggested below focuses on "how" the music is organized - the music theory question. (The "why" is normally the province of musicology.)
The purpose of an exploratory analysis of a piece of music is to create a quick overview of what you know and understand when you listen to the piece or look at a notated version of it. The analysis outlined below assumes that you have chosen a piece of music that you consider "good" and are mainly trying to understand how it is put together. This type of analysis is particularly useful for anyone who would like to compose, arrange, or improvise pieces in the same genre and style. If you understand how good composers, song-writers, improvisers, and arrangers have organized their music, it becomes much easier for you to put together good music of your own. Music-theory analysis is also useful for musicians who are preparing to perform a piece, write a critique of it, studying music theory or researching the history or sociology of music.
A "complete" analysis, explaining every aspect of every sound in a piece, would probably be impossible. A very thorough analysis of even a short piece of music would require much effort and is likely to produce much that is obvious, irrelevant, or uninteresting. Most thoughtful and useful analyses focus on the most interesting and instructive aspects of the piece. A musician who is experienced at analysis can quickly and easily identify the most important aspects of the piece. Musicians who are less experienced may want to do a formal preliminary analysis, such as the one outlined below. The point of creating this overview analysis is to identify the aspects of the piece that you do not yet understand and believe to be instructive and interesting. The rest of your analysis can then focus on understanding those aspects of the piece.