Sinfonia is the Italian (Spanish, and also Portuguese) word for symphony (see that article for etymology). In music Sinfonia has however some specific meanings and connotations, that are understood when the word sinfonia is used outside the realm of Latin-based languages:
A specific form of such kind of preluding piece, in the early 18th century, was the three-movement sinfonia which became the standard type of overture to an Italian opera. Most of the time these pieces were in D major (for maximizing open-string resonance on string instruments), opening and ending with a fast movement, with a slow movement in the middle. Examples of this type of Italian sinfonia are the numerous three-movement opera overtures by Alessandro Scarlatti, all archetypical Italian overtures.
In France however overtures had always been one-movement preluding pieces, usually in a A-B-A form, where the "A" sections had a slow tempo with a stately (double)dotted rhytm, while the "B" middle section was comparatively fluent and fast. This musical form became known as the French overture. By the time this type of overture was adapted by German composers like Bach and Handel from the early 18th century on, it could be as well the preluding movement of a (dance) suite, in which case overture was sometimes used as a synonym for the entire suite (e.g. Bach's French Overture, BWV 831).
Most of Handel's operas and oratorios start with the French type of overture movement, even if he occasionally calls such movement a sinfonia (as he did for the Messiah, actually calling it a ). But Handel would use the Italian type of orchestral prelude/interlude too, for instance the Introduzione to the cantate Delirio amoroso, HWV 99. Also the instrumental featuring in the Messiah did not so much derive from French examples. An interesting anecdote is that when Mozart made a German version of the Messiah, some 30 years after Handel's death, he changed the name of the opening Sinfony to Ouvertüre, but more or less did away with its French characteristics: he softened the dotted rhythm of the "A" section with some more flowing horn melodies, and by speeding it up a bit also made it less distinct from the "B" section: the result is that the "A" part appears as not much more than a moderate preamble to a "fast" symphonic movement (the "B" section).
In the mean while, also from the early 18th century on, the 3-movement Italian type of sinfonia had started to lead a life on its own: it could be composed as an independent concerto-like piece (without soloists however). For instance Vivaldi composed as well 3-movement independent sinfonias, not so different from some of his string concertos, as well as composing similar sinfonia preludes for his operas.
Bach sometimes used the term sinfonia in the then-antiquated meaning of an instrumental single-movement piece, e.g. for the keyboard Inventions and Sinfonias BWV 787-801, using a three-voice polyphonic style. Note that in 20th century, publishers started to publish these sinfonias as "Three-Part Inventions", where "Part" is an independently flowing melody ("voice", but in the instrumental meaning) in a single-movement work.
If Bach opened a vocal work with one or more separate instrumental movements (which was all in all not so often), he would usually call such piece a sinfonia or alternatively a sonata. For the sinfonias the style would be rather Italian (also for the single-movement ones) than French:
Both Handel and Bach used the French type of overture to start their orchestral suites. For suites they composed for a solo instrument there often was no preluding movement. If there was, that opening movement would usually be either an Overture/Ouverture (in that case always referring to the French style), or otherwise a Prelude/Praeludium. The style of such preludes was less defined but would often emulate the style of a fast movement of an Italian sinfonia.
As the 18th century progressed, the usual name for an instrumental prelude to a vocal/theatrical work would settle on overture. Although such overtures would generally be one-movement pieces, they were no longer in the French style, but rather adapted the Italian preluding sinfonia, for instance a loud, triadic, motto-type leading motif, a reprise preceded with minimal thematic development, and an overall mood of expectation rather than resolution.
The idea of the Italian 3-movement sinfonia as an independent orchestral composition lived on too: the earliest symphonies of Haydn and Mozart were composed in this format. Mozart also composed divertimentos in the Italian sinfonia format, with some ambiguity whether such divertimentos were indeed intended as independent instrumental compositions, or rather as instrumental interludes (for theatre productions etc).
But then Haydn made the Italian sinfonia/non-solistic concerto and the French type of overture/suite meet again: he took the three movements of a sinfonia, and inserted a fourth between the two last movements of the Italian model. That additional movement was a menuet, which had until then only been an almost obligatory movement of a suite. He also took some characteristics of the French style overture movement, as well as of what was the sonata in those days, amongst others the possibility to start the first movement of such four-movement composition with a slow introductory passage. But the resulting composition was no longer called a "sinfonia" (at least not outside Italy and Spain): the symphony was born.
Examples of such "sinfonias" composed after the classical era include: