As Apollos pointed out, here are some of the very earliest ones (i.e. ones that were made before AD 250):
Rylands Library Papyrus P52 (ca. early-mid 2nd century AD) - contains John 18:31-33 (recto) and 19:37-38 (verso).
Papyrus 90 (late 2nd century AD) - contains John 18:36-19:1 (recto) and 19:1-7 (verso)
[
Papyrus 98 (
http://en.wikipedia.org/wiki/Papyrus_98) (late 2nd century AD) - contains Revelation 1:13-2:1
Papyrus 104 (late 2nd century AD) - contains Matthew 21:34-37 (recto), 43 and 45? (verso)
[
Papyrus 4 (
http://en.wikipedia.org/wiki/Papyrus_4) (late 2nd-early 3rd century AD) - contains (extensive parts of) Luke 1-6. Along with P75, it is one of the earliest manuscripts of the Gospel of Luke.
Papyrus 66 (ca. AD 200) - contains John 1:1-6:11, 6:35b-14:26, 29-30; 15:2-26; 16:2-4, 6-7; 16:10-20:20, 22-23; 20:25-21:9, 12, 17.
Papyri 64 and 67, aka the Magdalen Papyrus (ca. AD 200) - contains Matthew 3; 5; 26.
Papyrus 32 (ca. AD 200) - contains Titus 1:11-15; 2:3-8
Papyrus 75 (ca. AD 175-225) - contains (extensive portions of) Luke 3:18-24:53 and John 1-15. One of the earliest manuscripts of the Gospel of Luke, along with Papyrus 4.
Papyrus 46 (ca. AD 175-225) - Surviving leaves (most of which are somewhat damaged) contain:
- Romans 5:17-6:3, 6:5-14, 8:15-25, 27-35, 8:37-9:32, 10:1-11, 11:24-33, 11:35-15:9, 15:11-end (with 16:25-27 following chapter 15!);
- Hebrews 1:1-9:16, 9:18-10:20, 10:22-30, 10:32-13:25;
- 1 Corinthians 1:1-9:2, 9:4-14:14, 14:16-15:15, 15:17-16:22;
- 2 Corinthians 1:1-11:10, 12-21, 11:23-13:13;
- Ephesians 1:1-2:7, 2:10-5:6, 5:8-6:6, 6:8-18, 20-24;
- Galatians 1:1-8, 1:10-2:9, 2:12-21, 3:2-29, 4:2-18, 4:20-5:17, 5:20-6:8, 6:10-18;
- Philippians 1:1, 1:5-15, 17-28, 1:30-2:12, 2:14-27, 2:29-3:8, 3:10-21, 4:2-12, 14-23;
- Colossians 1:1-2, 5-13, 16-24, 1:27-2:19, 2:23-3:11, 3:13-24, 4:3-12, 16-18;
- and 1 Thessalonians 1:1, 1:9-2:3, 5:5-9, 23-28, in that order.
Papyrus 45 (c. AD 250) - Contains Matthew 20-21, 25-26; Mark 4-9, 11-12; Luke 6-7, 9-14; John 4-5, 10-11; and Acts 4-17
As for the earliest complete text of the NT we have; that is, one that has all the books bound under one volume, the credit goes to both the 4th-century (ca. AD 330-360)
Codex Sinaiticus, the only uncial manuscript with a complete (surviving) text of the New Testament, and
Codex Vaticanus (c. 325–350) - which however contains only the Gospels, Acts, the General Epistles, the Pauline Epistles and Hebrews (up to 9:14). 1 and 2 Timothy; Titus; Philemon; and Revelation were missing and was supplemented by a 15th century minuscule hand (some even think that maybe Revelation was not included after all).