I only know the British and Irish history of the topic, but here it is:

Up until the nineteenth century, public libraries didn't really exist. The Library at the British Museum (now the British Library) has been in existence since 1753, but is a consulting rather than lending library, and it was difficult to get hold of a reader's card. Only Thomas Plume's library in Maldon (established in 1704) let visitors in to borrow books. Occasionally parish libraries would be open to the public (select visitors, at least) but these tended to have the books chained to the stacks, as in the Francis Trigge library. A rare example of a public institution is Chetham's Library in Manchester (1653) but here, again, the books were chained.

Wealthy people already had access to libraries, of course. Here in Belfast, for example, there has been a subscribing library since 1788. The genesis of the type that you're talking about, however, was in working men's associations of the early nineteenth century. The first of these sprung up in the early 1820s in Glasgow and Liverpool, but the most prominent was the London Mechanics Institute, which was founded in 1823. These institutions aimed to educate the adult men of the working classes, and included lessons and a lending library. An interesting example that I've done a bit of research on is the Morpeth Mechanical and Scientific Institute, which was set up in 1825. The Radical local businessman and philosopher Robert Blakey was approached by two Newcastle businessmen, John Fenwick and Emerson Chanley, who were interested in forming such an institution in Morpeth. Blakey was initially hesitant, because he felt that the general public would be resistant to working class education and its association with radical politics; nevertheless he agreed to assist in its foundation. He produced a pamphlet on the value of education, which he distributed around Morpeth, and called a meeting of wealthy townsfolk to obtain funding. Blakey obtained the necessary funding and was entrusted with funds to buy books in Edinburgh, eventually stocking the institute with hundreds. Members paid a small fee, and were allowed to borrow books from the library.

With the popularity of these institutions, an increase in literacy, and expansion of the franchise after the Great Reform Act, pressure was placed on politicians to provide a similar service to the general public (and keep these newly literate workers doing something productive with their free time!). This went through a few abortive attempts before the Public Libraries Act in 1850, but it took another Act in 1866 to make library provision practical, and even then progress was very slow.

You can judge its success by the amount of libraries provided by philanthropists. The most famous of these was Andrew Carnegie, who funded thousands of public libraries across the trans-Atlantic world in the late eighteenth and early nineteenth centuries, but there are dozens of other philanthropists who built libraries across the UK during this time. Even in a large city like Glasgow, the provision was poor enough that the businessman Stephen Mitchell left a bequest in his will to found a reference library. Here in Belfast, the lovely Central Library was established in 1888, and was widely applauded by the local community. Belfast was, for most of the nineteenth century, a staunchly Presbyterian town, which placed a great emphasis on literacy. Across the water, once the initial reluctance to support the working classes in case they got ideas above their station had passed, they were very well received.

Sources:

The Intellectual Life of the British Working Classes by Jonathan Rose Memoirs by Robert Blakey George Birkbeck: pioneer of adult education by Thomas Kelly