Here's an example which might help.
Consider the sound(s) spelled with the letter T in the following words: toe, stow, pity, hit. Are they the same sound?
On the physical, concrete level, they clearly aren't. The t in toe is pronounced with aspiration -- a puff of air -- which is mostly absent in stow. The t in pity sounds like neither of those, but is flapped (at least for most American English speakers) so that it actually sounds rather like a Spanish r. And the t in hit is usually not released at all, and is often pronounced as a glottal stop, without any contact at all between the tongue and the roof of the mouth.
On a more abstract level, though, these are clearly felt by English speakers as being in some sense "the same sound". This can be shown by the fact that they alternate predictably in words formed from the same root: change hit to hitter and the t becomes a flap.
The first, concrete level is the phonetic level, and the four kinds of t sounds are called allophones. They're notated with square brackets: [tʰ t ɾ ʔ]. The second, more abstract level is the phonological level, and the single abstract t sound that the allophones are all realizations of is called a phoneme. It's notated with slashes: /t/.
Phonemes are "contrastive" whereas allophones are not, because if you replace one allophone with another (for example if you say hit with the t of toe) you don't get a different word, while if you replace one phoneme with another (for example change /t/ in hit to /p/) you get a different word (hip).
Allophones are (at least sometimes) "contextually determined" because the choice between allophones depends on the position of the sound in the word: for example, a word-final t gets pronounced as a glottal stop, a word-initial t gets aspirated, etc.
This is an example from phonology, but as the passage you quote says, there are similar patterns in morphology (word formation). For example, the plural suffix (abstract "morpheme") is realized with different concrete "allomorphs" in different words, e.g. cat-s vs. ox-en.