In linguistics, the brackets <...> are used to represent written letters (e.g. <cat>); the brackets [...] are used to represent sounds (e.g. [k], [ae], [t] or [kaet]); the brackets /.../ are used to represent phonemes (e.g. /k/, /a/, /e/, /t/ or /kaet/).
Written letters: <cat>
Here's the difference between phonetic representation and phonemic representation:
Notice the difference in representation between [m] and /n/. When linguists transcribe Japanese (i.e. write down the sound of a language), they hear [kombanwa] "Good evening", so they write down that [k] and [o] and [m] and so on as sounds of Japanese. Then, once enough sounds are transcribed, linguists start looking at the distribution of each sound (i.e. where they occur in words). The reason they do that is because sounds change depending on their environment. That they change has to do with the physiology of the mouth and the throat. The more sounds share similar pronunciation features, the easier their articulation.
In the case of Japanese, linguists notice that [m] always occurs before , and, moreover, if is deleted, then [n] appears, like this,
kon-ban-wa => [kombaenwa]
kon => [kon]
The sound change rule or phonological rule is that dental /n/ is pronounced as labial [m] before a labial sound .
/n/ is the true sound; [m] is its variant. The brackets slanted brackets /.../ tell us the true nature of the sound. The square brackets [...] tell us how the sound is pronounced.
Phonemes (/.../) are not sounds. They represent the true natural of a sound ([..]) before it is spoken.