The letter A began long ago in Egypt as the image of an ox's head. Later it was aleph at the beginning of the Phonetician alphabet, the world's first full alphabet. The Greeks, needing vowel symbols, used it for alpha. The Romans used it as A in their alphabet and passed it to the British. Much later the English started using the Roman alphabet instead of their own runic script.
The word 'a' (indefinite article) is a Middle English lazy form of the Old English word 'an' meaning 'one'. Between the 12th and 14th centuries the letter 'n' began to disappear before consonants. The word 'an' was used before words beginning with the letter 'h' until about 1600. The word 'an' is still used in front of words beginning with the letter 'h', by some writers, in a few unusual cases, like 'historic' and 'herb'. The 'h' in herb is silent in American English. An hotel (silent h as in the French) was more formal in England until recently.
The vowels (a,e,i, o, u, oo and ignoring y) all have more than one sound for the letter. They at least have a short and long sound. There are four sounds for the letter 'a'. There is the short sound /æ/ as in 'cat'. The long sound /eɪ/ as in 'ape'. The broad sound is the ah-sound /ɑ:/ as in 'father'. The fourth sound is the neutral schwa sound /ə/ which is most often represented by the letter 'a'. For example: About, America, Russia and of course China.
The long a-sound is sometimes spelt without the letter a, as in ei, ey, and eigh. Just to be confusing, ei can be a long e-sound, as in 'ceiling', a long a-sound as in 'hey' or a short e as in 'foreign'. Also ey can be a long e-sound, as in 'honey', a long a-sound as in 'obey' or a long i-sound as in 'eye'. There are very few root words, but many compounds, with eigh and they are leftover from Old English. They are: eight, weight, height, neighbour, neigh (the sound a horse makes). The exception is freight, a word borrowed from Dutch in the 13th century. Dutch is of course closely related to Old English.
The letter 'a' is found in many blends. It helps the aw/al sound, as in law, all, stall and ball. Also it is in ar-sound as in hard, barn, tar and far.
Different English accents sometimes switch around which words get which a-sound. In some southern British accents bath and grass are pronounced with the broad-a. This is different from American English and most northern UK English varieties, in which these words are pronounced with the short-a. So expect some inconsistencies.