I was reading about this recently, I can't remember where. I might try and find it. In the meantime:

Rome in the Late Republic had a relatively large number of literate people and most of the upper class would have had a small library in their house(s). There absolutely were bookshops, known as taberna librarii in Latin. They would have sold exclusively scrolls, as the 'book' in the modern form (codex) didn't become popular until a couple of centuries later. Most tabernae librarii were in the Argiletum, a district of Rome leading from the forum to the Subura (the Subura being the dangerous and poverty-stricken area in between two of the hills North-East of the city centre). There were also travelling booksellers.

As to why books were written, it depends on the book. The wealthiest Romans were surrounded by a troupe of poets whom they commissioned to write books of poetry - either epic poems such as the Aeneid, or books of smaller poems, such as those written by the likes of Ovid and Horace. And many wealthy Romans wrote books themselves, on fashionable topics such as philosophy or history, and presumably financed the publishing themselves.

Some Roman historians, such as Sallust, were ex-politicians (Sallust 's career came to a somewhat ignominious end when he was accused of corruption in the administration of his province) who betook themselves to history writing as a project for their retirement - Sallust in particular used his history as a way of criticising his former political opponents.

Others such as Cicero published during their working lifetimes. Cicero published his legal speeches starting from an early point in order to publicise himself and aid in his career. In fact one of his speeches, 'in Verrem' (against Verres) was published before it was even given, with the result that Verres fled Italy rather than continue with the trial since the speech had been circulated beforehand and was very damning. He also had many of his letters to his friends published as books since they were on subjects he thought people would find interesting (actually I'm not sure his friends didn't have a lot of them published on his behalf for the same reason, as a tribute to him).

There were no prose novels in ancient Rome (well, there are a few of examples of prose fiction, mostly Greek rather than Latin) so most books were either poetry, philosophy, history or other forms of non-fiction, including more practical manuals on things like medicine or architecture and farming (or, in the odd case of Virgil's Georgics, a cross between poetry and farming).

Many books in Rome would have been in Greek, since the upper classes were usually fluent in Greek. I don't know to what extent other languages were represented, probably barely at all since most foreigners in Rome were slaves and most slaves from places which didn't speak Greek or Latin were likely illiterate. Maybe you could find a couple of works in Aramaic or Demotic, probably of a religious nature. In fact one would imagine there were people selling Hebrew religious scriptures.

Sources:

Horace Odes Bk. 1 (Cambridge Greek and Latin Classics) by R. Mayer, various pages
Cicero Pro Caelio (ditto) by A. Dyck, various pages
Literature in the Roman world, O. Taplin Ed. chapters 2, 3 and 4