Who was the second Pope after St Peter?

Answer 1

No one. Peter wasn't a Pope. He was an Apostle. The Pope is the bishop of Rome.

Answer 2

Here is a list of all the Popes:
Peter (32-67)
Linus (67-76)
Anacletus (Cletus) (76-88)
Clement I (88-97)
Evaristus (97-105)
Alexander I (105-115)
Sixtus I (115-125) -- also called Xystus I
Telesphorus (125-136)
Hyginus (136-140)
Pius I (140-155)
Anicetus (155-166)
Soter (166-175)
Eleutherius (175-189)
Victor I (189-199)
Zephyrinus (199-217)
Callistus I (217-22)
Urban I (222-30)
Pontain (230-35)
Anterus (235-36)
Fabian (236-50)
Cornelius (251-53)
Lucius I (253-54)
Stephen I (254-257)
Sixtus II (257-258)
Dionysius (260-268)
Felix I (269-274)
Eutychian (275-283)
Caius (283-296) -- also called Gaius
Marcellinus (296-304)
Marcellus I (308-309)
Eusebius (309 or 310)
Miltiades (311-14)
Sylvester I (314-35)
Marcus (336)
Julius I (337-52)
Liberius (352-66)
Damasus I (366-83)
Siricius (384-99)
Anastasius I (399-401)
Innocent I (401-17)
Zosimus (417-18)
Boniface I (418-22)
Celestine I (422-32)
Sixtus III (432-40)
Leo I (the Great) (440-61)
Hilarius (461-68)
Simplicius (468-83)
Felix III (II) (483-92)
Gelasius I (492-96)
Anastasius II (496-98)
Symmachus (498-514)
Hormisdas (514-23)
John I (523-26)
Felix IV (III) (526-30)
Boniface II (530-32)
John II (533-35)
Agapetus I (535-36) -- also called Agapitus I
Silverius (536-37)
Vigilius (537-55)
Pelagius I (556-61)
John III (561-74)
Benedict I (575-79)
Pelagius II (579-90)
Gregory I (the Great) (590-604)
Sabinian (604-606)
Boniface III (607)
Boniface IV (608-15)
Deusdedit (Adeodatus I) (615-18)
Boniface V (619-25)
Honorius I (625-38)
Severinus (640)
John IV (640-42)
Theodore I (642-49)
Martin I (649-55)
Eugene I (655-57)
Vitalian (657-72)
Adeodatus (II) (672-76)
Donus (676-78)
Agatho (678-81)
Leo II (682-83)
Benedict II (684-85)
John V (685-86)
Conon (686-87)
Sergius I (687-701)
John VI (701-05)
John VII (705-07)
Sisinnius (708)
Constantine (708-15)
Gregory II (715-31)
Gregory III (731-41)
Zachary (741-52)
Stephen II (752) -- Because he died before being consecrated, some lists (including the Vatican's official list) omit him.
Stephen III (752-57)
Paul I (757-67)
Stephen IV (767-72)
Adrian I (772-95)
Leo III (795-816)
Stephen V (816-17)
Paschal I (817-24)
Eugene II (824-27)
Valentine (827)
Gregory IV (827-44)
Sergius II (844-47)
Leo IV (847-55)
Benedict III (855-58)
Nicholas I (the Great) (858-67)
Adrian II (867-72)
John VIII (872-82)
Marinus I (882-84)
Adrian III (884-85)
Stephen VI (885-91)
Formosus (891-96)
Boniface VI (896)
Stephen VII (896-97)
Romanus (897)
Theodore II (897)
John IX (898-900)
Benedict IV (900-03)
Leo V (903)
Sergius III (904-11)
Anastasius III (911-13)
Lando (913-14)
John X (914-28)
Leo VI (928)
Stephen VIII (929-31)
John XI (931-35)
Leo VII (936-39)
Stephen IX (939-42)
Marinus II (942-46)
Agapetus II (946-55)
John XII (955-63)
Leo VIII (963-64)
Benedict V (964)
John XIII (965-72)
Benedict VI (973-74)
Benedict VII (974-83)
John XIV (983-84)
John XV (985-96)
Gregory V (996-99)
Sylvester II (999-1003)
John XVII (1003)
John XVIII (1003-09)
Sergius IV (1009-12)
Benedict VIII (1012-24)
John XIX (1024-32)
Benedict IX (1032-45)
Sylvester III (1045) -- Considered by some to be an antipope
Benedict IX (1045)
Gregory VI (1045-46)
Clement II (1046-47)
Benedict IX (1047-48)
Damasus II (1048)
Leo IX (1049-54)
Victor II (1055-57)
Stephen X (1057-58)
Nicholas II (1058-61)
Alexander II (1061-73)
Gregory VII (1073-85)
Victor III (1086-87)
Urban II (1088-99)
Paschal II (1099-1118)
Gelasius II (1118-19)
Callistus II (1119-24)
Honorius II (1124-30)
Innocent II (1130-43)
Celestine II (1143-44)
Lucius II (1144-45)
Eugene III (1145-53)
Anastasius IV (1153-54)
Adrian IV (1154-59)
Alexander III (1159-81)
Lucius III (1181-85)
Urban III (1185-87)
Gregory VIII (1187)
Clement III (1187-91)
Celestine III (1191-98)
Innocent III (1198-1216)
Honorius III (1216-27)
Gregory IX (1227-41)
Celestine IV (1241)
Innocent IV (1243-54)
Alexander IV (1254-61)
Urban IV (1261-64)
Clement IV (1265-68)
Gregory X (1271-76)
Innocent V (1276)
Adrian V (1276)
John XXI (1276-77)
Nicholas III (1277-80)
Martin IV (1281-85)
Honorius IV (1285-87)
Nicholas IV (1288-92)
Celestine V (1294)
Boniface VIII (1294-1303)
Benedict XI (1303-04)
Clement V (1305-14)
John XXII (1316-34)
Benedict XII (1334-42)
Clement VI (1342-52)
Innocent VI (1352-62)
Urban V (1362-70)
Gregory XI (1370-78)
Urban VI (1378-89)
Boniface IX (1389-1404)
Innocent VII (1404-06)
Gregory XII (1406-15)
Martin V (1417-31)
Eugene IV (1431-47)
Nicholas V (1447-55)
Callistus III (1455-58)
Pius II (1458-64)
Paul II (1464-71)
Sixtus IV (1471-84)
Innocent VIII (1484-92)
Alexander VI (1492-1503)
Pius III (1503)
Julius II (1503-13)
Leo X (1513-21)
Adrian VI (1522-23)
Clement VII (1523-34)
Paul III (1534-49)
Julius III (1550-55)
Marcellus II (1555)
Paul IV (1555-59)
Pius IV (1559-65)
Pius V (1566-72)
Gregory XIII (1572-85)
Sixtus V (1585-90)
Urban VII (1590)
Gregory XIV (1590-91)
Innocent IX (1591)
Clement VIII (1592-1605)
Leo XI (1605)
Paul V (1605-21)
Gregory XV (1621-23)
Urban VIII (1623-44)
Innocent X (1644-55)
Alexander VII (1655-67)
Clement IX (1667-69)
Clement X (1670-76)
Innocent XI (1676-89)
Alexander VIII (1689-91)
Innocent XII (1691-1700)
Clement XI (1700-21)
Innocent XIII (1721-24)
Benedict XIII (1724-30)
Clement XII (1730-40)
Benedict XIV (1740-58)
Clement XIII (1758-69)
Clement XIV (1769-74)
Pius VI (1775-99)
Pius VII (1800-23)
Leo XII (1823-29)
Pius VIII (1829-30)
Gregory XVI (1831-46)
Pius IX (1846-78)
Leo XIII (1878-1903)
Pius X (1903-14)
Benedict XV (1914-22)
Pius XI (1922-39)
Pius XII (1939-58)
John XXIII (1958-63)
Paul VI (1963-78)
John Paul I (1978)
John Paul II (1978-2005)
Benedict XVI (2005—)

With love in Christ.

Answer 3

It didn't work that way. In Rome in the earliest centuries, there was a collegial leadership group of overseers (episkopoi) of the Roman community. There wasn't just one "pope," but the term pope wasn't used then. Rome didn't get a mono-episkopos (one bishop of Rome) until the middle of the second century.
In Christian tradition, though, the second "pope" was Linus.

Answer 4

St. Linus (appox 67 - 79 c.e.)