See this explained in the notes at Mark 1:21-39.

Came down to Capernaum - Which it is likely he made his ordinary place of residence from this time. See on Matthew 4:13 (note).

And came down to Capernaum,.... Which was, as Dr. Lightfoot thinks, sixteen miles or more from Nazareth; and Christ may be said to "come down" to it, because of the situation of Nazareth, which was upon an hill:

a city of Galilee: of lower Galilee, near the sea of Galilee:

and taught them on the sabbath days: that is, he went into the synagogue at Capernaum, on the sabbath days, whenever he was there, and taught the inhabitants, explained some passage or other in the Old Testament, as he had done at Nazareth and instructed them in the doctrines of the Gospel.

Taught (ἠν διδάσκων)

Correctly, as Rev., was teaching. The finite verb and participle denoting continuance.

On the Sabbath-days (τοῖς σάββασιν)

Rev., day. The word is often used in the plural form for the single day, as in Luke 4:16; probably after the analogy of plural names of festivals, as τὰ ἄζυμα, the feast of unleavened bread; τὰ γενέσια, the birth-day; or perhaps following the Aramaic plural.

And came down to Capernaum, a city of Galilee, and taught them on the sabbath days.

4:31 Came down to Capernaum. Nazareth was on the hills; Capernaum in the deep depression of the Sea of Galilee.

4:31 He came down to Capernaum - And dwelt there, entirely quitting his abode at Nazareth. Mr 1:21.

31. down to Capernaum-It lay on the Sea of Galilee (Mt 4:13), whereas Nazareth lay high.

4:31-44 Christ's preaching much affected the people; and a working power went with it to the consciences of men. These miracles showed Christ to be a controller and conqueror of Satan, a healer of diseases. Where Christ gives a new life, in recovery from sickness, it should be a new life, spent more than ever in his service, to his glory. Our business should be to spread abroad Christ's fame in every place, to beseech him in behalf of those diseased in body or mind, and to use our influence in bringing sinners to him, that his hands may be laid upon them for their healing. He cast the devils out of many who were possessed. We were not sent into this world to live to ourselves only, but to glorify God, and to do good in our generation. The people sought him, and came unto him. A desert is no desert, if we are with Christ there. He will continue with us, by his word and Spirit, and extend the same blessings to other nations, till, throughout the earth, the servants and worshippers of Satan are brought to acknowledge him as the Christ, the Son of God, and to find redemption through his blood, even the forgiveness of sins.