What does the Catholic church teach about fate of non-Catholic christians? From what I've heard, The Roman Catholic Church claims to be the only representative authority of God on Earth. Does it indeed claim that, and if so what does it say happens to members of non-Catholic churches when they die?
The Catholic Church is very explicit in its teachings that "all salvation comes from Christ the Head through the Church" (CCC 846).  The only way to enter the Church and become part of the Body of Christ is through baptism.  Thus, baptism is necessary for salvation (CCC 1257).

This is not meant to be interpreted as only Catholics can attain salvation.  Rather, the Church teaches that "anyone, even a non-baptized person, with the required intention, can baptize, by using the Trinitarian baptismal formula (CCC 1256)."  Thus, anyone baptized with water and "in the name of the Father, and of the Son, and of the Holy Spirit" is part of the Mystical Body of Christ and can attain salvation. To the best of my knowledge, this includes most non-Catholic Christians. (Note there are other forms of baptism recognized, baptism by blood and baptism by desire, but those seem out of scope for this question.)

Now Baptism isn't the only requirement for salvation.  One must also die free from mortal sin (CCC 1033). For a sin to be mortal, that is to be grave enough to cut us off from the life of God, three conditions must be met: "Mortal sin is sin whose object is grave matter and which is also committed with full knowledge and deliberate consent." (CCC 1857). This full knowledge includes both knowledge that it is a sin, and that it is a grave matter (CCC 1859).  Thus a sinner who through no fault of his own, believes he is not sinning does not commit a mortal sin.

If one does fall into a state of mortal sin, then normally the sacrament of reconciliation is necessary (CCC 1856) to restore or "reconcile" one with God.  However, it is recognized that a perfect contrition can also attain this forgiveness (CCC 1452).

In conclusion, the Catholic Church teaches that the sacraments of the Church are the surest and easiest way to reach salvation, but it is still possible for other members of the Body of Christ to enter Heaven.