What are the four essential attributes of the church as found in the Nicenoconstantinopolitan creed? How are we to understand each of them?If you recall from the Nicene Creed we recite in Mass, at the end we declare "I believe in one, holy, Catholic, and apostolic Church." You called them the "essential attributes" of the Church, but they are more commonly referred to by the Fathers and by theologians as the "four marks of the Church."
A mark is something that gives you distinction. It is a characteristic that identifies you from among a group of things. Thus, Jesus' Church, the Church He founded, is one, holy, Catholic, and apostolic. These are the characteristics that identify His Church. If your church does not have these characteristics, then it wasn't founded by Christ.
The Catechism does an excellent exposition of all four of these marks. See nos. 811-870. I'll provide my own summary remarks from what I am able to recall about each one.
The Church is "one", first of all, because She is united under the Pope, the Vicar of Christ. The Pope is a symbol of unity because anyone who does not acknowledge his authority is not one with us in the fullest sense. The Church is also one in Her teaching. Preserved down through the ages is the one deposit of faith, without contradiction, and our common adherence to this deposit binds us together.
The Church is "holy" because of the great fountain of grace that is made available to Her in the Sacraments. Through them, the Lord presents His Bride, the Church, to the Father without spot or wrinkle. The Church is also holy because of the overabundance of saints that She has produced, holy men and women who act as a great cloud of witnesses, interceeding for us with perfect prayers to the Father. Finally, the Church is holy because Her doctrine is without the stain of error. It is immutable and incorruptible.
The Church is "Catholic" because she is universal. The word "Catholic" means "universal." The Catholic Church is a worldwide Church. It encompasses men of every culture, race, nationality, and language. Furthermore, every day the Church seeks to live out Christ's command to "make disciples of all nations" and in Her great missionary work in foreign lands She follows in the footsteps of the Apostles.
Finally, the Church is "apostolic." This is, one, because the authority first granted to the apostles is preserved through their successors, the bishops. Secondly, through this succession, and through the charism of infallibility, the Church preserves and safeguards the teaching of the Apostles. Thus, not only her shepherds but also her very teaching is "apostolic." Lastly, the Church is apostolic because she identifies herself as "one who is sent." Such is the definition of the Greek word apostolos, from which we derive our word "apostle." Like I said before, the Church takes the Great Commission very seriously.
I just hit a few main points. Much more can be said, which is why you need to read the Catechism articles that mentioned. I hope this helps.