There are three meanings of the word "sequence" in music, two of them specifically technical and one more general. The musical genre of the sequence is a chant or hymn, usually composed to couplets in Latin. The best-known example is the Christmas carol "Adeste Fideles", the English version is "Oh, Come All Ye Faithful". Here is an earlier example, "Veni Creator Spiritus":
UPDATE: I am informed by a commentator (who obviously knows more about Catholic liturgy than I do) that the example of sequence I want is actually Veni sancte Spiritus:
The second meaning of the term "sequence" comes from music theory and refers to a harmonic or melodic pattern repeated at different pitches. Here, from the Wikipedia article is a simple melodic sequence of a four-note scale segment repeated ascending one step each time:
There are also harmonic sequences that were developed in the 17th century and by Vivaldi's time had come to be used extensively. The effect for the listener is a bit like being on a Ferris wheel, with the feeling of rotating through space. Here is an example of the most common type, the descending fifths sequence, in which the bass line descends a fifth each time:
There are several other varieties of varying complexity and composers have displayed considerable ingenuity in their use. If you want examples, you will find them in nearly every quick movement from the Baroque and Classical eras. Rather than repeat myself, I will refer you to two posts I have put up on sequence previously here and here.
But those two uses aside, there is a general sense in which we can use the word "sequence" in talking about music. Like theater and dance, music is a "time-art" meaning that it takes place in time. Musical events are ordered in a dramatic sequence. Now in the 20th century many attempts were made to suppress or eliminate this aspect of music as part of the general modernist project of experimentation. Some composers, like John Cage, eliminated the dramatic sequence entirely by choosing notes by chance procedures so that no logical sequence was possible. He also wrote a piece consisting only of silence that achieves the same end. Other composers made pieces consisting of lengthy drones which also tend to downplay any sequential drama. Steve Reich and Philip Glass wrote pieces in which the events are so extended that the sensation is almost that there is no temporal progression--one feels trapped in a single moment. New Age meditation music seeks the same feeling.
But all these experiments, to my mind at least, really demonstrate that the unique strength and power of music is that very thing: the dramatic sequence. All attempts to avoid or eliminate it tend to be boring as music, fascinating though they may be from the point of view of metaphysics or meditation.
The great masters of dramatic impetus in music are found in the 18th and early 19th centuries. Before then, composers were still discovering how to assemble musical structures that drove forward, taking the listener on a kind of journey. As we move into the 19th century, the whole dynamic tends to slow down as something called the "romantic trance" is developed. But for that fairly brief time between 1700 and 1830, music achieved a remarkable ability to grab the listener, take them on a stimulating and multi-faceted journey and deposit them back home, fulfilled and amused. That's quite a feat, given that the mechanism is just a few compression waves in the air.
It is sequence that achieves it all. A sequence of short and long notes within an underlying pulse gives us rhythm. A sequence of different pitches building to a climax and tapering to a finish gives us the melodic phrase and the organization of harmony into a principal key and excursions into related harmonies, near or far, gives us the large structure. Master all of them and you can construct a piece of music.
Well, enough blather. Let's listen. Here is an interesting early example by C. P. E. Bach, who was the bridge between the somewhat more static structures of the Baroque, and the dynamism of the Classical style. Here is the Orchestra della Svizzera Italiana conducted by Ton Koopman playing the Symphony No.1 in D major, H.663 Wq.183 by C. P. E. Bach. And yes, there are some pretty good examples of the harmonic sequence here and there.