The term fellowship has no specific definition given to it within the New Testament, however, we are called into fellowship with one another (1 John 1:3), with Jesus (1 Cor. 1:9), with the Father (1 John 1:3), and with the Holy Spirit (2 Cor. 13:14). Fellowship implies sharing common interests, desires, and motivations. Fellowship requires that time be spent with another communicating, caring, etc. It carries with it a hint of intimacy. As Christians, we fellowship with one another because of our position in Christ and because we are all redeemed and share an intimate personal knowledge of Jesus. We share a common belief (Acts 2:42), hope (Heb. 11:39-40), and need (2 Cor. 8:1-15).
The Greek word for fellowship is koinonia. This word is also translated communion in 1 Cor. 10:16 in the KJV. This is where we get the term the communion supper.