The concerto, as a style, first appeared in the latter part of the Baroque period.
There were two styles
This form is said to have its roots in many of the antiphonal works of composers such as Gabrielli earlier in the Renaissance period. Two opposing groups, so often heard in the instrumental and choral music of Gabrielli, were replaced by:
Many great composers of the period wrote concertos in this style, in particular Corelli, J S Bach, Handel and Vivaldi.
These works could have anything from three to five movements, or sections, of contrasting speeds and styles.

First movements were frequently in ritornello form, a form in which the whole orchestra, the ripieno, alternated with sections played by soloists, the concertino.
To help clarify this in your mind, listen to a short example of the ripieno or tutti at the start of the movement playing the ritornello, the name given to the opening section and the main theme of the movement.
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Now listen to the concertino playing in an episode, the name given to a section played by the soloists and be sure to also identify the continuo accompanying the soloists in the background.
Listen to the start of the first movement of J S Bach's Brandenburg Concerto No.2 and listen to the alternating use of the ripieno, concertino and the continuo playing continuously in the background.

Now listen to a whole movement from a concerto grosso by Handel and listen carefully for the contrasting texture of the sections played by the ripieno and the concertino. Listen for the two violins of the concertino, the frequent use of ascending and descending sequences, repetition, a counter melody on a cello, a move to a minor key in the middle and a pause just before the final perfect cadence.
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It must be remembered that most music at this time was written for a purpose and for performance at a particular occasion. At this time the composition of the concertino was influenced by the quality and number of really good players the composer had available for a particular performance.

It must come as no surprise to find running parallel to the concerto grosso, the development of the first solo concertos by the same composers.
These were generally in three movements, fast, slow and fast.
In many of the solo concerto movements the soloist would first of all play along with the strings of the orchestra and then embellish the melody line by adding a more flamboyant version above the orchestra.
A perfect example of this can be heard in the Violin Concerto in E Major by J S Bach. Listen to the opening of this and notice how all the strings play together and then the solo violin takes over the melody and decorates it with a more complicated line, often using arpeggios.

Vivaldi wrote many solo concertos and in particular for oboe, flute and bassoon. Listen to part of the first movement of his Concerto in C Minor. In particular listen for many of the trademarks of Vivaldi's music, the use of repetition, sequence, simple harmony, pedal and ornamentation.
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As well as writing concerto grosso, Handel also wrote solo concertos, and among his most famous are the ones for organ.
Listen to this short example in which the organ can be clearly heard imitating a well-known bird, the other name given to the concerto.