Origins
The word 'letter' carries two meanings that have traveled together since antiquity: a character of the alphabet and a written message. Both senses descend from Latin 'littera' (a letter of the alphabet), whose plural 'litterae' meant variously 'letters' (of the alphabet), 'a letter' (an epistle), 'literature,' and 'learning' in general. This polysemy was inherited wholesale by Old French 'letre' and passed into Middle English around 1200.
The ultimate origin of Latin 'littera' is uncertain. One hypothesis connects it to Latin 'linere' (to smear, to spread), suggesting that the original concept was the smearing of ink or pigment onto a surface to form characters. Another traces it to an Etruscan intermediary, since the Romans received their alphabet from the Etruscans, who had received theirs from the Greeks. The uncertainty is appropriate for a word whose referent — the alphabetic letter — is itself borrowed technology, passed from Phoenicians to Greeks to Etruscans to Romans.
The singular/plural distinction in Latin was semantically productive. Singular 'littera' meant one character: A, B, C. Plural 'litterae' could mean the alphabet as a whole, a written document, a body of writing, or the quality of being educated. A 'homo litterātus' (lettered man) was not someone who knew his ABCs but someone steeped in literature and learning. This is the origin of English 'literate' (able to read and write, but with overtones of education) and 'illiterate' (unable to read, but with overtones of ignorance).
Latin Roots
The derivative vocabulary is extensive. 'Literal' (from Latin 'litterālis') means 'pertaining to the letter' — following the exact letters of a text, not interpreting figuratively. 'Literary' (from 'litterārius') means 'pertaining to literature.' 'Alliteration' (from 'ad-' + 'littera') means 'to the letter' — the repetition of initial letters or sounds. And 'obliterate' (from 'oblitterāre') means 'to strike out the letters,' 'to erase' — originally a scribal term for crossing out text, which expanded to mean destroying anything so thoroughly that no trace remains.
German took a different path entirely. The German word for an alphabetic letter is 'Buchstabe,' literally 'beech-staff' — a compound of 'Buche' (beech) and 'Stab' (staff, stick), referring to the beech-wood staves on which runes were carved. This gives German two distinct words where English has one: 'Buchstabe' for a character of the alphabet and 'Brief' (from Latin 'brevis,' short) for a written message.
The postal sense of 'letter' — a written communication sent to a recipient — developed naturally from the Latin plural 'litterae' (a written document). Roman 'litterae' could be official dispatches, personal correspondence, or legal documents. The English 'letter' retained this full range, though by the modern period the word came to denote primarily personal correspondence sent through a postal system. The rise of email has not killed the metaphor: we still speak of 'electronic letters' in many languages, and the at-sign (@) in email addresses was originally a scribal abbreviation used in written letters.