Here is a very brief explanation.
[ ] - represents a phone, an actual sound.
/ / - represents a phoneme, a meaningful unit of sound within the sound system of a language.
< > - represents a grapheme, a unit of spelling.
// // is a specialized subject. I won't address it.
Examples (in American English):
The graphemes < ph > and < f > have the sound [f] in English.
The phoneme / p / has three allophones in English: [pʰ] (aspirated [p]) as in < pat >, [p] (unaspirated [p]) as in < spat >, and [p̚] (unreleased [p]) as in < tap >.
With broad phonetic transcription, /p/ is represented by [p].
< pat > is pronounced [pæt], is pronounced [spæt], and < tap > is pronounced [tæp].
With narrow phonetic transcription, /p/ is represented by [pʰ], [p], and [p̚].
< pat > is pronounced [pʰæt̚], is pronounced [spæt̚], and < tap > is pronounced [tʰæp̚].
Also, here is a link to a useful IPA keyboard site.