In simplest terms, a Christian disciple is a committed follower of Jesus Christ. In the Bible, "disciple" is synonymous with "Christian." As one source explains:
The characteristic name for those who gathered around Jesus during his ministry was "disciple." He was the teacher or master; they were his disciples (mathētai), a term involving too much personal attachment and commitment to be adequately rendered by "pupil." The name was carried over into Acts, where it frequently has the general sense of Christian (cf. Acts 14:21). The use of the term in Acts for those who had no acquaintance with Jesus during the days of his flesh serves as a reminder that the relationship of subsequent generations of Christians to the exalted Christ is not essentially different from that enjoyed by those who walked with him on the earth. It further indicates that the Lord's solemn sayings regarding the nature and cost of discipleship must be given their full weight in the construction of a doctrine of the Christian life (cf. Luke 6:40; 14:25-33). However common the term may be in Acts, it disappears completely from the rest of the NT. Apparently "disciple" was no more adequate as a permanent title for Christians than "teacher" or "rabbi" was for Jesus, whose lordship was now fully revealed and understood. (Evangelical Dictionary of Theology) (quoted verbatim)
Today we speak of disciples and discipleship 1) as a means of identifying true, committed Christians, and 2) as shorthand for the process of growing and maturing in our Christian walk and witness. True, saving faith in Jesus Christ involves much more than mere mental assent to certain facts. It involves placing our complete trust in Christ, and then living for God rather than self. Faith and faithfulness are inseparable: salvation joins us both to Christ and to fellow Christians, and the Holy Spirit "nurtures the initial response of faith to the gospel message into a continuing life of joyful praise and obedience to God (Rom. 1:5; 16:26), and loving service to the neighbour (Gal. 5:6, 13–14; 6:1–2)." (New Dictionary of Biblical Theology).
True, committed discipleship is very costly and, in fact, involves nothing less than a radical rethinking of what life is all about. As Jesus put it: "'If anyone comes to Me, and does not hate his own father and mother and wife and children and brothers and sisters, yes, and even his own life, he cannot be My disciple. Whoever does not carry his own cross and come after Me cannot be My disciple'" (Luke 14:26-27, NASB). This means that: