The First Viennese School is a name mostly used to refer to three composers of the Classical period in Western art music in late-18th-century Vienna: Wolfgang Amadeus Mozart, Joseph Haydn and Ludwig van Beethoven. (Franz Schubert is occasionally added to the list).
In German speaking countries, the term Wiener Klassik (lit. Viennese classical era/art) is used. That term is often more broadly applied to the Classical era in music as a whole, as a means to distinguish it from other periods that are colloquially referred to as classical, namely Baroque and Romantic music.
The term "Viennese School" was first used by Austrian musicologist Raphael Georg Kiesewetter, in 1834, although he only counted Haydn and Mozart as members of the school. Other writers followed suit and eventually Beethoven was added to the list. The designation "first" is added today to avoid confusion with the Second Viennese School.
With the exception of Schubert, these composers certainly knew each other, with Haydn and Mozart even being occasional chamber-music partners. But they were not engaged in a formal effort to associate with 20th-century schools, such as the Second Viennese School, or Les Six. Nor is there any evidence that one composer was "schooled" by another, in the way that Berg and Webern were taught by Schoenberg. Although it is true that Beethoven for a time received lessons from Haydn.
Attempts to extend the First Viennese School to include such later figures as Anton Bruckner, Johannes Brahms, and Gustav Mahler are merely journalistic, and never encountered in academic musicology.