The minuet is a graceful, courtly dance of French origin in 3/4 time. It was adopted into the suite in the late 17th century and placed between the sarabande and the gigue. Minuets usually come in pairs and since each is a binary dance, the form is AABB (first minuet) AABB (second minuet) and finally, AB (first minuet again, this time without repeats). So the whole form is ABA with A being the first minuet and B the second. The second minuet, in the music of Lully, who popularized the dance, is often scored for a trio of instruments such as two oboes and a bassoon and so the whole structure was often called "minuet and trio". In this form, it was the only dance of the baroque suite to survive into the classical era where it became a standard part of the symphony. A lively version became the classical scherzo such as was used by Haydn and Beethoven.
The gigue is a lively dance originating in Britain where it survives to this day as the Irish and Scottish jig. In the 17th century it was adopted by composers on the continent and found a place as the last dance in the suite. It is usually in 3/8 or one of the compound meters such as 6/8, 9/8 or 12/8, though there are a couple of exceptions by Bach. I talk about one of them in this post. In Bach's hands the gigue can get rather complex at times with each half becoming semi-fugal and the second half inverting the subject of the first half. The gigue to the Suite in A major is not one of these, but a rather straightforward gigue in 6/8 time.
Time signatures like these, called "compound" are a simplification of the Medieval mensural notation. Mensuration is the relationship between any note value and the next smaller note-value. In modern notation this relationship is always duple: every smaller note is half the value of the larger note. Less abstractly, every half breaks down into two quarter notes, every quarter into two eighths and so on. In mensural notation, there are always two options: duple or triple. The breve could break down into two or three semibreves and this was indicated by the time signature. A circle indicated a triple division and a half-circle, a duple division. This latter time signature survives into modern notation and music teachers who don't know their history tell their students the half circle (equivalent to 4/4 time) is a "C" standing for "common time". Actually, it stands for "tempus imperfectus cum prolatio imperfecta"! Triple was "perfect" and duple "imperfect" to Medieval musicians.
In modern notation we have replaced these complexities with two things: meters where the subdivision is triple, not duple, we call "compound" and we think of the beat as a dotted note (divisible in three). In all other cases, we use the triplet sign of a bracket with a "3" in the middle. Inelegant, but simpler than the Medieval system.
Between 1717 and 1723 Bach was employed by Prince Leopold of Cöthen (now spelled Köthen) and this is where he wrote the six cello suites. In the clip I have included a picture of the St. Agnus Church in Köthen, which Bach attended, the first page of the manuscript of the cello suites in the hand of his wife, Anna Magdalena Bach, the opening of the first minuet and some pictures of myself. Accompanying the gigue are photos of the Bach monument in Leipzig, just outside the Thomaskirche, where he is buried, the Bach Museum and archive, also in Leipzig and the first page of my edition of the gigue.