... And Boy Are My Arms Tired       - Kill enemies with unarmed weapons
... And Not A Drop To Drink          - Drink Water Items
... Or Are You Just Happy To See Me? - Explode enemy pants
A Fistful of Hollars                 - Assault people with an unarmed weapon
A Little Critical                    - Get Critical Hits With Pistols
Animal Control                       - Kill mutated animals
Artful Pocketer                      - Successfully pickpocket people
Beam (Weapon) Me Up                  - Kill enemies with energy weapons
Bug Stomper                          - Kill mutated insects
Click                                - Kill enemies with automatic weapons
Condition Critical                   - Get critical hits with rifles
Crafty                               - Craft items
Critical Reaction                    - Get critical hits with energy weapons
Day Tripper                          - Use addictive chems
Demise of the Machines               - Kill robots
Determines Survivalist               - Eat pieces of food
Devourer of Nations                  - Cannibalize corpses
Fast Times                           - Consume Jet
Find Vault 75                        - Discover Vault 75
Find Vault 81                        - Discover Vault 81
Find Vault 95                        - Discover Vault 95
Find Vault 114                       - Discover Vault 114
Find Vault 118                       - Discover Vault 118
Find Vault Arcturus 1                - Discover Vault Arcturus 1
Free Radical                         - Use Rad-Away
Healthy Glow                         - Successfully hack terminals
He Moves in Mysterious Ways          - Witness Mysterious Stranger appearances
I've Got Shotgun                     - Kill enemies with shotguns
Just Point and Click                 - Kill enemies with heavy weaponry
Lead Dealer                          - Kill enemies with guns
Lethal Weapons                       - Get critical hits with unarmed weapons
Low Tech Hacking                     - Successfully pick locks
Master of the Commonwealth           - Discover locations
Maybe That's a Lie                   - Become addicted to a substance
New Commonwealth Samurai             - Kill enemies with melee weapons
Radical                              - Consume Rad-X
Sir Talks-a-lot                      - Pass speech checks
Stimpaddict                          - Use Stimpaks
Up Close                             - Kill enemies with melee weapons

Challenges are ordered in-game as they are on the wiki page.
FALLOUT NEW VEGAS CHALLENGE LIST