To make (both) ends meet means to earn just enough money to live on.
It is first recorded in The History of the Worthies of England (1662), by the Church of England clergyman Thomas Fuller (1607/8-61). The author wrote the following about the English Protestant leader Edmund Grindal (1519-83) – in the original text, to put off his clothes and went to bed on the one hand, make both ends meet and lapped over on the other hand, are in italics because Fuller is quoting existing idioms:
Being really blind more with grief then age, (dying at sixty four) he was willing ‘to put off his clothes’ before he ‘went to bed’, and in his life time to resigne his place to Doctor Whitgiff, who refused such acceptance thereof. And the Queen, commiserating his condition, was graciously pleased to say, that, As She had made him, so he should die an Arch bishop, as he did July 6. 1583.
Worldly wealth he cared not for, desiring onely to ‘make both ends meet’; and as for that little that ‘lapped over’, he gave it to pious uses in both Universities, and the founding of a fair Free-school at Saint Bees, the place of his nativity.
The phrase is from tailoring or dressmaking, and refers to the amount of material needed to make a piece of clothing reach round the body, so that its two ends meet. This is what Thomas Fuller seemed to imply with “that little that lapped over” in the above-mentioned passage.
This explanation seems to be supported by the second-earliest occurrence of the phrase, in A New Dictionary of the Terms Ancient and Modern of the Canting Crew (1699), by “B. E. Gent.”:
Ends […] Tis good to make both Ends meet, or to cut your Coat according to your Cloth.
In the phrase, both ends denotes the extremities of the year (that is, the beginning and the end), the verb meet means to agree or tally, so that the whole means to keep one’s finances, income and expenditure, in balance throughout the year.
This corresponds to the French expression joindre les deux bouts, to join both ends, which used to end with de l’année, of the year. The following definition is from the 1st edition (1694) of the Dictionnaire de l’Académie française:
Il ne sçauroit joindre un bout de l’année avec l’autre, pour dire, Son bien ne luy suffit pas pour aller jusqu’au bout de l’an.
He cannot join one end of the year to the other, to say, His goods are not sufficient for him to go until the end of the year.
However, this explanation seems to correspond to a late extension of the English phrase, which is to make the (two) ends of the year meet, first recorded in The Adventures of Roderick Random (1748), by the Scottish author Tobias George Smollett (1721-71):
This facetious person was a school-master, whose income being small, he was fain to keep a glass of good liquor for the entertainment of passengers, by which means he made shift to make the two ends of the year meet.
“I am,” said he, “a single man, have a considerable annuity on which I live according to my own inclination, and make the ends of the year meet very comfortably.”
Old China, by the English author Charles Lamb (1775-1834), first published in The London Magazine of March 1823, contains:
It is mighty pleasant at the end of the year to make all meet—and much ado we used to have every Thirty-first Night of December to account for our exceedings—many a long face did you make over your puzzled accounts, and in contriving to make it out how we had spent so much—or that we had not spent so much—or that it was impossible we should spend so much next year—and still we found our slender capital decreasing.
Those two explanations are not mutually exclusive: the two earliest attestations seem to indicate that the original image was the amount of cloth necessary to make a garment reach round the body, while the later uses seem to show that the phrase was reinterpreted as referring to bookkeeping.