Some simple definitions:

Phoneme	An ideal sound unit with a complete set of articulatory gestures The basic theoretical unit for describing how speech conveys linguistic meaning. For English, there are about 42 phonemes. Types of phonemes: vowels, semivowels, dipthongs, and consonants.
Phonemics	The study of abstract units and their relationships in a language.
Phone	The actual sounds that are produced in speaking (for example, "d" in letter pronounced "l e d er").
Allophones	The collection of all minor variants of a given sound ("t" in eight versus "t" in "top") Monophones, Biphones, Triphones - sequences of one, two, and three phones. Most often used to describe acoustic models.

Three branches of phonetics:

Articulatory phonetics	manner in which the speech sounds are produced by the articulators of the vocal system.
Acoustic Phonetics	sounds of speech through the analysis of the speech waveform and spectrum.
Auditory phonetics	studies the perceptual response to speech sounds as reflected in listener trials.

Issues: Broad phonemic transcriptions vs. narrow phonetic transcriptions