The Westron or Common Speech is the closest thing to a universal Language, at least at the time during the War of the Ring. Westron is a translation of the original name Adûni (cf. Adûnaic Adûn "west"), and "Common Speech" translates the Westron term Sôval Phârë, of identical meaning. In Sindarin the language was called Annúnaid (Westron), or Falathren (Shore-language).
The Westron speech is derived from the Adûnaic tongue of Númenor, and originated as a creole language on the western coastlands of the continent of Middle-earth, when the Númenoreans established trade outposts and forts there. From there, it spread to most of the westlands, with the notable exception of Mordor.
After the Downfall of Númenor, the Faithful Númenoreans neglected their 'unfaithful' language in favor to Elvish, letting Adûnaic as spoken in Middle-earth to change and evolve chaotically among the Middle Men. But later it was enriched and softened under Elvish influence. It became the lingua franca of most explored regions of (north-west) Middle-earth, known at least as far east as Esgaroth, as the language of trade and diplomacy. For example under King Thengel of Rohan (who had lived in Gondor for many years before taking the throne), the Common Speech started being used as the language of the court instead of their native Rohirric (thus Theoden, Eomer, and Eowyn are all functionally bilingual in Rohirric and Common Speech) "though not all thought this a good thing..."
Westron has both "deferential" pronouns, and "familiar" pronouns, but Hobbitish no longer possesses a deferential pronoun. Westron had deferential pronouns for the second person (and sometimes the third person) but this had fallen out of use in Hobbitish.
This lack of a deferential pronoun and universal use of the familiar pronoun is what Gondorians are referring to when they repeatedly remark that Hobbit-speech sounds strange.
According to Tolkien's fiction, Westron was the language spoken and understood by the protaginists of The Hobbit and The Lord of the Rings. Many names of characters and places, in the book's "reality", were in Westron.
However Tolkien, mentions that Westron was presented as having been completely replaced by English in the text. This had certain important implications: first of all, proper names with derivations understandable or evident to speakers of Westron had been translated, to preserve the effect to the English reader. Thus, names like Baggins, Bagshot Row, Peregrin, Rivendell et cetera, are not the actual names as spoken by the characters but are presented as translations.
Of course, outside the fictional context of the story, it is clear that there was no such "translation": the English names came first and the "original" forms in Westron or other languages were devised by Tolkien later.
Rivendell ("cloven valley") was actually called Karningul, and Bag End was actually called Labin-nec, after Labingi, the real form of Baggins. In some cases the explanations became quite involved, such as the river Brandywine (Sindarin Baranduin, "golden-brown river") was actually called Branda-nîn, a punning Westron name meaning "border-water", which was later punned again as Bralda-hîm meaning "heady ale".
This logic went one step further by also presenting all languages akin to Westron in languages related to English, so that their "understandability" by the protagonists be simulated to the English reader. For further information cf. the article about Mannish.
This utter replacement of Westron by English was taken so far that some sources that should give actual Westron have been turned to English too. For instance, in Moria, an illustration of the runic text on Balin's gravestone is given. The text is said to be written in both Khuzdûl and Westron. But while the first part of the inscription seems to really be a bit of Khuzdûl, the second part is actually plain English, just written in cirth.
The corpus of Westron is small; several of the Westron forms given above were not published in Tolkien's lifetime. Tolkien never worked out Westron to the same extent as Quenya and Sindarin or even Adûnaic.