See the account of the raising of Jairus' daughter, and the healing of the woman with an issue of blood, fully explained in the notes at Matthew 9:18-26.

And, (g) behold, there cometh one of the rulers of the synagogue, Jairus by name; and when he saw him, he fell at his feet,
(g) The whole company did not assemble without any structure, but in every synagogue there were certain men who governed the people.

And behold, there cometh one of the rulers of the synagogue,.... Having heard of his return, and where he was; See Gill on Matthew 9:18.
Jairus by name; in Hebrew "Jair": and Jerom says (n), it signifies "enlightening", or "enlightened": deriving it from "to be light": and it is no doubt a Jewish name, since he was a ruler of the synagogue; and besides, it is often mentioned in the Old Testament, and particularly in Esther 2:5, where, in the Septuagint, it is read, Jairus. Matthew makes no mention of his name; but both Mark and Luke do, Mark 5:22.
And when he saw him, he fell at his feet: as soon as he came into his presence; though he was a person of such authority; yet having heard much of the doctrine and miracles of Christ, and believing him to be a great prophet, and man of God; though he might not know that he was the Messiah, and truly God, threw himself at his feet; and, as Matthew says, "worshipped him", Matthew 9:18; showed great reverence and respect unto him, gave him homage, at least in a civil way, though he might not adore him as God.
(n) De Hebr. Nominibus in Luc.

And, behold, there cometh one of the rulers of the synagogue--of which class there were but few who believed in Jesus (John 7:48). One would suppose from this that the ruler had been with the multitude on the shore, anxiously awaiting the return of Jesus, and immediately on His arrival had accosted Him as here related. But Matthew (Matthew 9:18) tells us that the ruler came to Him while He was in the act of speaking at His own table on the subject of fasting; and as we must suppose that this converted publican ought to know what took place on that memorable occasion when he made a feast to his Lord, we conclude that here the right order is indicated by the First Evangelist alone.
Jairus by name--or "Jaeirus." It is the same name as Jair, in the Old Testament (Numbers 32:41; Judges 10:3; Esther 2:5).
and when he saw him, he fell at his feet--in Matthew (Matthew 9:18), "worshipped Him." The meaning is the same in both.

One of the rulers of the synagogue. The office of ruler in the synagogue was somewhat similar to that of elder in a Christian congregation.

One of the rulers of the synagogue - To regulate the affairs of every synagogue, there was a council of grave men. Over these was a president, who was termed the ruler of the synagogue. Sometimes there was no more than one ruler in a synagogue. Matthew 9:18; Luke 8:41.