Cardinals can belong to one of three orders:
What were the ranks of Cardinals Richelieu and Mazarin?
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At the moment of his election (1641), it seems that Mazarin was in minor orders - so called "lay cardinal". After that, there seems to be little consensus and pretty much no primary sources, but if anything, he was a cardinal-priest.
By the process of elimination, he was a cardinal-priest:
He was definitely not a cardinal-deacon. From "The Cardinals of the Holy Roman Church - Biographical Dictionary"
Cardinalate. Created cardinal in the consistory of December 16, 1641; never received the red hat and the title or deaconry.
He was also not a cardinal-bishop.
From Wiki definition: "the title of cardinal bishop only means that the cardinal in question holds the title of one of the 'suburbicarian' sees — they include the Dean of the College of Cardinals".
There is no information of any see he held (and the Cardinals list above lists one for every other cardinal); and he explicitly turned down a bishopric offer from Richelieu:
Richelieu, in spite of his fondness and admiration for Mazarin, was loath to crown his career so early; he offered a bishopric worth 30,000 écus a year. Mazarin, who aspired to more, for his part, turned it aside amiably.
As a separate confirmation, "The Catholic encyclopedia: an international work of referenc..." edited by Charles George Herbermann in 1911 states:
M. Loiseleur, who has made a careful study of the problem, believes that Mazarin was never married; it is certain that he retained the title and insignia of a cardinal until his death; probably he was even a cardinal-priest, though he never visited Rome after his elevation to the purple and seems never to have received the hat. And in any case he held the title of Bishop of Metz from 1653 to 1658.
Another evidence is from http://www.gcatholic.com/hierarchy/data/cardU08.htm (the web site seems to be some sort of Catholic Encyclopedia, but can't say how authoritative it is). There, Mazarin is listed as:
former Cardinal-Priest with no Title assigned former Abbot Ordinary of Cluny (France) Born: 1602.07.14 (Italy) Ordained Priest: Created Cardinal: 1641.12.16 Cardinal-Priest with no Title assigned (1641.12.16 – 1661.03.09) Bishop of Metz (France) (1652 – 1658) Abbot Ordinary of Cluny (France) (1654 – 1661.03.09)
Another reference: http://www.csun.edu/~hcfll004/SV1644.html
"Cardinals not attending:" (SEDE VACANTE July 20, 1644—September 15, 1644)
Jules Raymond Mazarin (aged 44) Cardinal Priest without titulus (arrived too late).
Not sure what the primary source is for "Cardinal Priest" piece but the bibliography for that page lists:
For the Conclave of 1644, see Giuseppe de Novaes, Elementi della storia de' sommi pontefici da San Pietro sino al ... Pio Papa VII third edition, Volume 10 (Roma 1822) 6-10. Alexis Francois Artaud de Montor, Histoire des souverains pontifes V (Paris 1851) 375-390. F. Petruccelli della Gattina, Histoire diplomatique des conclaves III (Paris 1865), pp. 96-129. Hermann Conring (editor), De electione Vrbani IIX et Innocentii X Pontificum Commentarii historici duo (Helmestadii: Henningus Mullerus 1651). Ernesta Chinazzi, Sede vacante per la morte del papa Urbano VIII Barberini e conclave di Innocenzo X Pamfili (agosto-settembre 1644) (Roma, 1904).
Henri Coville, Étude sur Mazarin et ses démêlés avec le Pape Innocent X. 1644-1648 ( Paris: Champion 1914).
According to Cardinal Richelieu's Wikipedia page he was a cardinal priest until December 4, 1642, the day of his death. Mazarin is difficult to find specific information on. According to his Wikipedia page, Jules Mazarin succeeded Richelieu. Since I cannot find any information on which kind of cardinal Mazarin was, I can only assume that he was a cardinal priest as well, like his mentor, Richelieu. Since Richelieu was chief minister to the monarch of France, and Mazarin was also, I hypothesize with confidence that he was, in fact, a cardinal priest.
What sort of Cardinals were Richelieu and Mazarin?
Let us start with Cardinal Mazarin.
He was a Cardinal Priest.
Birth. September 1, 1605, Pescina, Abruzzi. Baptized on that same day, received the name Alessandro. Son of Pietro Mazzarino and Ortenzia Buffalini. The family lived in Rome but his father administered the benefice of a relative in Pescina. Brother of Cardinal Jules Mazarin (1641). He is also listed as Michele Mazzarini.
Education. Entered the Order of Preachers (Dominicans) at the convent of S. Maria sopra Minerva, Rome, when he was 15 years old; took the name Michel. Completed his studies in Bologna.
Priesthood. Ordained (no information found). Professor of philosophy and theology in houses of study of his order. Successfully complete a delicate mission in Venice given to him by the master general of his order. Provincial of the Dominican province of Puglia, and of the Roman province, 1641. In October 1642, Pope Urban VIII named him vicar general of his order and deposed Niccolò Ridolfi, master general. He was elected master general in a chapter held in Genoa and presided over by him. Pope Innocent X considered the motives for the deposition of the master general to be insufficient. Opposed to his election were the Spaniards, the Flemish and the Germans, who then elected Fr. Tommaso di Roccamara. Fr. Mazarin resigned in the interest of peace and to avoid a schism within the order. Master of the Apostolic Palace, February 3, 1643.
Episcopate. Elected archbishop of Aix-en-Provence, July 10, 1645. Consecrated, Sunday July 23, 1645, Dominican church of S. Maria sopra Minerva, Rome, by Cardinal Girolamo Grimaldi, assisted by Alfonso Sacrati, former bishop of Comacchio, and by Ranuzio Scotti, bishop of Borgo San Donnino. Returned to Rome from his see in 1647 (1).
Cardinalate. Created cardinal priest, with dispensation because he had a brother in the Sacred College of Cardinals, in the consistory of October 7, 1647; received the red hat and the title of S. Cecilia, December 16, 1647. Named viceroy of Catalonia, he entered Barcelona in January 1648 but had to leave shortly, in July of the same year, because of ill health. Returned to Rome as French minister in July 1648.
Death. August 31 (2), 1648, Rome. Exposed and buried in the church of S. Maria sopra Minerva, Rome.
Bibliography. Cardella, Lorenzo. Memorie storiche de' cardinali della Santa Romana Chiesa. 9 vols. Rome : Stamperia Pagliarini, 1793, VII, 72-74; Combaluzier, Fernand. "Sacres épiscopaux à Rome de 1565 à 1662. Analyse intégrale du Ms. «Miscellanea XIII, 33» des Archives Vaticanes." Sacris Eruduri, XVIII (1967-1968), p. 229.
(1) According to Cardella, Memorie storiche de' cardinali della Santa Romana Chiesa, VII, 73, apparently to address some affairs of the French king before the Holy See but in reality because he did not adapt himself to the customs and manners of the French which he found odioso, e insoffrible.
(2) This is according to Hierarchia Catholic Medii et Recentioris Aevi, IV, 29. Dictionnaire des cardinaux, col. 1230, indicates that he died on September 2, 1648 at 41 years of age.
Armand Jean du Plessis, Duke of Richelieu is more well known as Cardinal Richelieu and was a Cardinal Priest.
Birth. 1582, Paris, France. Of a noble family. Second son of François du Plessis de Richelieu and Suzanne de la Porte de Vezins. Brother of Cardinal Armand-Jean du Plessis de Richelieu (1622).
Education. Received his early education under the care of his mother in the castle of his paternal grandparents where, together with his two brothers and two sisters, he had moved after the death of the father in 1590. Collège de Navarre, Paris, 1594-1601. Obtained a doctorate in theology in order to occupy the bishopric of Luçon that corresponded almost as inheritance to his family. When he declined, his brother Armand-Jean took his place as bishop of that see.
Early life. Declined promotion to the see of Luçon offered by King Henri IV of France in 1602; entered the Carthusian order, 1602; professed, 1605.
Priesthood. Ordained (no information found). Prior of the Carthusian monastery of Bompas. In 1626, his brother the cardinal arranged for his promotion to the episcopate.
Episcopate. Elected archbishop of Aix-en-Provence, April 27, 1626. Consecrated, June 21, 1626, Carthusian church, Paris, by Guillaume d'Hughes, archbishop of Embrun, assisted by Claude de Rueil, bishop of Bayonne, and by Nicolas Sanguin, bishop of Senlis. Transferred to the metropolitan see of Lyon, November 27, 1628. Abbot commendatario of Saint-Michel, 1628.
Cardinalate. Created cardinal priest in the consistory of November 19, 1629, with dispensation for having a brother in the Sacred College of Cardinals. Named by King Louis XIII in 1631, Grand Aumonier of France. Abbot commendatario of Saint-Etienne de Caen, 1632; and of Saint-Victor de Marsella or La Chaise-Dieu. Commander of the Order of Saint-Esprit, 1633. Provisor of La Sorbonne University, Paris. Dean of Saint Martin de Tours. Received the red hat and the title of SS. Trinità al Monte Pincio, June 4, 1635. During the plague epidemic in Lyon in 1638, he himself assisted in the nursing of the sick. In June 1644, as grand aumonier, he presided over the funeral of King Louis XIII. Participated in the conclave of 1644, which elected Pope Innocent X. He was widely appreciated for his humility and his merciful dedication to the poor. The relationship between the two cardinal brothers was not tension-free despite their close friendship. He introduced chocolate in France and it appears that his excessive consumption was his only vice.
Death. March 24, 1653 (1), of dropsy, Lyon. Buried, in accordance to his wishes, in the chapel of the Virgin Mary in the church of the Hôpital de la Charité of Lyon. When the hospital was demolished, his remains were transferred to the Primatial Cathedral of Lyon and reburied in its crypt. It was he who wrote his own epitaph: Je suis né pauvre, j'ai fait vœu de pauvreté, j'ai vécu pauvre, je meurs pauvre, je veux être enterré au milieu des pauvres.
Bibliography. Bergin, Joseph. The making of the French Episcopate, 1589-1661. New Haven; London : Yale University Press, 1996, ch. IV, "Biographical dictionary," p. 691; Cardella, Lorenzo. Memorie storiche de' cardinali della Santa Romana Chiesa. 9 vols. Rome : Stamperia Pagliarini, 1793, VI, 291-292; Chapeau, O.S.B. André and Fernand Combaluzier, C.M. Épiscopologe français des temps modernes, 1592-1973. Paris : Letouzey et Ané, 1974, p. 279-280; Deloche, M. Un frère de Richelieu inconnu. Paris : 1935; Gauchat, Patritium. Hierarchia Catholica Medii et Recentioris Aevi. Volumen IV (1592-1667). Münich : Sumptibus et Typis Librariae Regensbergianae, 1935; reprint, Padua : Il Messagero di S. Antonio, 1967, pp. 22, 50 and 226.
(1) This is according to Gauchat, Hierarchia Catholica Medii et Recentioris Aevi, IV, 22, 50 and 226; Chapeau, Épiscopologe français des temps modernes, 1592-1973, p. 279-280, says that he died on March 23, 1653 in Lyon.