The Merriam-Webster online dictionary defines a fellowship as a "community of interest, activity, feeling, or experience" and "a company of equals or friends." As a verb, to fellowship simply means to spend time with that community or company.
God designed His Church to be a fellowship—not just a community, but a family of brothers and sisters who share God's ideals and beliefs (Ephesians 2:19-22 Ephesians 2:19-22  Now therefore you are no more strangers and foreigners, but fellow citizens with the saints, and of the household of God;
 And are built on the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone;
 In whom all the building fitly framed together grows to an holy temple in the Lord:
 In whom you also are built together for an habitation of God through the Spirit.
American King James Version×). Our Church brethren can help us improve ourselves—for "as iron sharpens iron, a friend sharpens a friend" (Proverbs 27:17 Proverbs 27:17Iron sharpens iron; so a man sharpens the countenance of his friend.
American King James Version×, New Living Translation).
The importance of fellowship in our lives is part of the reason that the people of God are commanded to "consider one another in order to stir up love and good works, not forsaking the assembling of ourselves together, as is the manner of some, but exhorting one another, and so much the more as you see the Day approaching" (Hebrews 10:24-25 Hebrews 10:24-25  And let us consider one another to provoke to love and to good works:
 Not forsaking the assembling of ourselves together, as the manner of some is; but exhorting one another: and so much the more, as you see the day approaching.
American King James Version×, emphasis added). Every week, on God's Sabbath day, Christians gather together both to hear the Word of God taught and to strengthen each other through fellowship.
Our conversations with our spiritual brothers and sister and the encouragement we share is beneficial to us, can help others and is pleasing to God.
For more information about the Church of God, please read our Bible study aid The Church Jesus Built.