For example p as in pin and p as in spin are allophones in the English language. English treats these as the same, but they are different. The latter is unaspirated: it sounds a little more like the b of English. The preceding s is the usual context for the unaspirated allophone. Chinese treats them differently and the latter is written as b in pinyin: thus they are not allophones in Chinese.
The speakers of a language are not usually aware of allophonic differences: to them a p is a p'\'. But take the unaspirated p out of context and they might hear it differently: a recording of spin with the s left out might be heard as bin'' by an English-speaker.
A phoneme itself is an abstract thing. Not all phonemes need have significantly different allophones, but there will always be minor differences in articulation from one piece of speech to the next. A phone is a sound that has a definite shape as a sound wave, and an allophone is a phone considered as a member of one phoneme. Speakers of a particular language perceive a phoneme as a single distinctive sound in that language.