What are each of the challenges?
On the main menu, there are a list of challenges you can attempt. Clicking one starts the game, presumably with differences depending on the challenge.
What is the locked challenge, and what is different about each challenge?
Let's answer with a little help from the wiki. Each challenge is played as Isaac and each one has a specific goal.
Dark was the Night: Gives you Curse of Darkness (you cannot see the minimap). Ends after killing Mom. Completing this challenge unlocks The Candle, a usable item with instant recharge that throws flames that damage anything on their path.
7 Years Bad Luck: Gives you Curse of the Lost (still unclear what this curse does). Ends after killing Mom. Completing this challenge unlocks the Lucky Toe, a trinket that gives you a chance of extra items when opening chests or "X" rocks.
Large Marge: Gives you Curse of the Labyrinth (extra large double levels). Ends at the first Depths boss. Unlocks the Burnt Penny, a trinket that grants a chance of dropping a bomb when you collect a penny
9 Deaths: Gives you the Dead Cat item (9 1-heart lives available). Ends after killing Mom. Unlocks Guppy's Tail, an item that, if combined with Guppy's Head and Guppy's Paw, changes your appearance into a flying cat (other effects still unknown)
Lord of the Flies: Claims to give you Undefined, but actually gives you Infestation and The Mulligan (the first produces attack flies from your body when hit, the second when hitting enemies). Ends at the Womb's first boss. Unlocks the Fish head, a trinket that has a similar effect to The Mulligan.
The Doctor's Revenge!: Gives you the Dr Fetus item and the Doctor's remote. Still unclear if it removes treasure rooms. Ends after killing Mom's heart. Completing the challenge unlocks the Mega Fetus a missile shooting Dr Fetus.
Meat 4 Evar!: Starts you out with lv. 4 Meat Boy. Ends after killing Mom's heart. Completing this challenge unlocks the Super Fan, a saw-looking item that increases all stats by 1, same as the Halo.
Spider Boy!: Gives you the Spider's Bite item, and the Mutant Spider. Ends after killing Mom's heart. Completing this challenge unlocks the Spider Butt, a usable item with a 2 room recharge that damages all enemies in the room and slows them.
Isaac Was Good Today: Gives you the Holy Grail and the Halo. Ends after killing Satan. Unlocks Counterfeit Penny, a trinket that has a chance to spawn an additional cent when collecting money.
The Purist: You start with no items, spacebar or passive. Treasure rooms are removed from the maps as well. Bosses and special events still can drop collectables. Ends after killing Isaac. Unlocks the Egg Sack, a long leg that follows you and stomps enemies. Unlocked by beating Isaac.
There are still things to discover, I'll keep the answer updated as soon as I find them.