- Phrase is the big, generic term that at its most general simply means the sections that music tends to fall into that are very roughly analogous to sentences or phrases in language--which is where the term came from. Phrases are typically four, eight or sixteen measures long. What distinguishes a phrase are things like the shape of the melody (rising then falling, for example), harmonic closure at the end, and rhythmic structure (typically long notes at the beginning and shorter notes at the end, or the other way around).
- Motif is a short melodic and/or rhythmic cell that is typically used as a structural device. The opening four notes of Beethoven's 5th Symphony are an iconic example. Those anapests and dactyls I keep finding in the Shostakovich quartets are another.
- Theme refers to a melodic unit that is used structurally. A good example would be the subject to a Bach fugue. In some theories of classical style, a number of different types of themes are distinguished by the phrase and harmonic structure (see William Caplan's books)
Now let's see what Wikipedia says.
That was pretty much what I said, even with some of the same examples, only with a lot more words. The problem with all these definitions, and why you don't need to spend a lot of time on them, is that they are very general and very abstract. Music, on the other hand, is very specific and concrete! So you can define 'phrase' however you like, but you either have to avoid any specifics or find that the very first phrase you look at won't fit the definition. Bummer!
Here is the important paragraph from the Wikipedia article on "phrase" that quotes a number of different definitions:
John D. White defines a phrase as, "the smallest musical unit which conveys a more or less complete musical thought. Phrases vary in length and are terminated at a point of full or partial repose, which is called a cadence." Edward Cone analyses the "typical musical phrase" as consisting of an "initial downbeat, a period of motion, and a point of arrival marked by a cadential downbeat". Charles Burkhart defines a phrase as "Any group of measures (including a group of one, or possibly even a fraction of one) that has some degree of structural completeness. What counts is the sense of completeness we hear in the pitches not the notation on the page. To be complete such a group must have an ending of some kind … . Phrases are delineated by the tonal functions of pitch. They are not created by slur or bylegato performance … . A phrase is not pitches only but also has a rhythmic dimension, and further, each phrase in a work contributes to that work's large rhythmic organization."
OK. So, what's a complete musical thought? What's a point of repose? What if you don't have a cadence? What if the phrase starts on an upbeat, not a downbeat? What does "structural completeness" mean? The thing is that all these definitions sort of work if you are talking about classical period music, but the further away from that period, the less they fit. In Shostakovich, for example, even though his music is certainly tonal, it doesn't have traditional cadences so any attempt to understand phrasing in his music has to rely on some other principle.
And don't even start with Gregorian chant...
Now let's have some music. How about some non-Gregorian chant? This is a Mozarabic chant: