Where did the hierarchical designations of priest, cardinal, bishop, and pope come from in the Catholic church? Protestantism typically recognizes offices that include pastors, elders, and deacons.  Another common belief is the priesthood of all believers, and thus there is no such designation for a particular office. 

Catholicism, by contrast, has priests, cardinals, bishops, and popes.  Where did these designations come from and when did each of them arise?  I know that the word "priest" does appear in the New Testament, but I am unaware of any words from which cardinal, bishop, or pope would be derived.

(I wonder if for the first few centuries, there were only priests and a pope and then over time other offices could have been added.  This is just speculation, though, as I am not too familiar with Catholic history.)
Actually cardinals, bishops and pope, all hold one and the same degree of Holy Orders in the Catholic Church. All of them are bishops.

Pope: Bishop of Rome is called Pope. As the bishop of Rome he is considered to be first amongst equals.Supremacy of Popes Pope just means Father/Papa

Cardinals: They are special bishopssee note 1 chosen by the Pope to help him in his work. Most important work of a cardinal is to chose the next pope. Contrary to popular belief, Cardinals are not the bosses of other bishops, as the Pope is not the boss of all Catholics. Catholics believe that a Bishop receives his authority from God not from pope. To be more specific Cardinal is NOT ecclesiastical posts. That is, Unlike a bishop when elevated as a Cardinal one does not gain any special spiritual privileges. The word Cardinal is based on the Latin cardo (hinge), meaning "principal" or "chief". The term was applied in this sense as early as the ninth century to the important priests of the diocese of Rome. Still today all cardinals are given charge of a Church in Rome.

Now the difference between a bishop and a priest, can be understood in terms of Jewish Priesthood. Bishops are a special type of priests called High Priests, just like in a Jewish tradition. Catholics do believe in the priesthood of all believers. The Catholic Church believes in three different types of priestsref:


The priesthood of all believers (1 Peter 2:5–9);
The ordained priesthood (Acts 14:23, Romans 15:16, Titus 1:5); and
The high priesthood of Jesus (Hebrews 3:1).


Catholic Encyclopedia explains the hierarchical position of the presbyterate as follows:


  The relation of the priest to the bishop and deacon may be briefly explained by stating that he is, as it were, the middle term between the two, being hierarchically the subordinate of the bishop and the superior of the deacon (cf. Council of Trent, Sess. XXVI, can. vi). While the pre-eminence of the bishop over the priest consists mainly in his power of ordination, that of the priest over the deacon is based on his power of consecrating and absolving (cf. Council of Trent, loc. cit., cap. iv; can. i and vii).ref


Notes:


There have been lay cardinals in the past, and some cardinals have chose not to get consecrated as bishop due to their old age.
The point I want to make clear is that the difference is only in the name for cardinals, bishops and popes. Of-course they have different temporal powers suitable to their rank, but nothing spiritual.