Phonetics is the study of the different speech sounds that can create speech and the way these are created by the various 'organs of speech' in the body including the tongue, palate, lips, pharynx, etc. Phonology is sometimes considered to be a part of grammar : it is the study of the structure of speech sounds and how these can be combined to create clarity of meaning in a sentence. Just as there are grammar 'rules' that apply to the syntax of a sentence and the morphology of words, there are, of course, phonological rules.
In even very early childhood, children are able to produce (i.e. articulate) the full range of sounds needed to create all of the words used in any language, yet as language acquisition progresses, those phonemes that do not apply to English words become forgotten so much so that in later life, if a second language is then learned, the pronunciation of non-English phonemes needs to be re-made this time at a wholly conscious level, as opposed to the ability to pronounce each English phoneme without thought. Even 'non-words' such as 'erm', 'uh?', etc. use English phonemes.
A phoneme is the smallest unit of speech-sound that can - by itself - change the meaning of a word or be a complete word in itself. There are more than 44 phonemes in the standard English dialect which is 18 more than the number of letters in the alphabet. Each of these phonemes is given a symbol so that the accurate pronunciation of any English word can be represented in writing. The extra sounds we have above the number of letters we have available in part explains the complexities of English spelling (i.e. orthography). Consider the word might, in which there are three phonemesm-ight-t (represented as m/aj/t using the International Phonetic Alphabet or IPA), changing just a single phoneme can completely change the meaning of this word, e.g. mate, m-a-te (represented as m/e/t phonetically).
Phonology also covers the study of important sound features such as rhythm, pitch, tone, melody, stress and intonation. These phonological language features are also referred to as the prosodic or suprasegmental features of language.