Daily challenges vary from day to day, but are basicaly divided in three parts:
Lockbreaker.
*Depending on the difficulty you must add puzzle pieces in a certain amount of moves to "unlock" them.
Easy and medium start more at the edges of a puzzle, while the harder ones start with different pieces with spaces between them.
*Bug catcher
You must "catch" the bug in a space while you block off the space. Easy and medium have less pieces, while the harder difficulties have more.
*Jig Saw Jammer
You get a certain amount of time to finish several puzzles and score a certain amount of points. Clicking in pieces at a fast rate gives you "chains" which is like the combo in most other games. You get certain puzzle shapes like zig-zag shapes, square shapes and regular puzzles.
Edit: You either watch a bonus commercial or you subscribe to Jigsaw Premium. (I believe it is 10 bucks for a year or 90 cents per month)
This will give you no ads and standard +60 or 90 seconds on the clock. (total time remaining gives you extra points. )
Lockbreaker is the easiest as you check for the edges which lets you finish the quickest. Just finish any challenge and you get this!