Sonata Allegro form was a development of the classical era.
It was named "Sonata Allgro" because the final Allegro movement of a Sonata was most often created in this form.
Sonata Allegro form can simplistically be described as an ABA form.
the first A section is called the Exposition, this section is repeated, actually making the overall form AABA.
the B section , or Development forllows the Exposition.
In the A section of Recapitulation, the A material returns relatively intact.
The sonata-allegro form is part of the Sonata Cycle, which is the blueprint of Symphonies, string quartets, and other works like sonatas in the classical period. It is also called the First movement form, as it is always used as the first movement in the sonata Cycle. The sonata-allegro form contains three components sections; Exposition, Development, and Recapitulation.
In the Exposition, theme 1 or theme group 1 is introduced in the tonic key, Haydn, a composer in the Classical period, was one of the unique composers who usually implemented a slow introduction before the actual introduction of theme group 1.
the Exposition is repeated.
In the Development section, themes from the Exposition is fragmented, inverted, and articulated to show the extreme possibilities of this section.
A Coda is Italian for tail.
"Sonata allegro" form would be more propely called sonata form, since a sonata form movement need not be allegro, it could be slower, such as Andante (walking pace).
One of the most famous examples is the first movement of Beethoven's fifth symphony in c minor. the form is in three main parts; exposition, development and recapitulation.
The exposition has a first subject and one or more second subjects.
the recapitulation is a more- or- less repeat of the exposition but with differences.