Blue's Clues is a game that Blue made up to communicate with Steve or Joe on what Blue wants to say. It starts by Blue putting a pawprint on the screen, and whatever the last question they asked Blue was is what they're trying to figure out by playing the game. But they can't have pawprints all over the house, much less the video camera, so before they play they have to get rid of the pawprint somehow. By wiping it off, forcing against it, verbal communication, etc. Then they go see Sidetable Drawer or Carmen to get their Handy Dandy Notebook or the kazoo which they draw pictures of the clues down or to play them in so they can remember them. Then, they sing the Blue's Clues Theme Song to go over how to play the game.
After that, they can start their search for clues. If they see a pawprint that Blue put on the screen earlier, whatever the pawprint is on, is a clue, which they draw in their notebook. After finding three clues, they sit in the Thinking Chair to try and figure it out. They combine the clues together to what they think is the answer to Blue's Clues. After they think they figured out the answer, they ask Blue if they're right. If she says they are, that means they just figured out Blue's Clues, which also means that the game is finished.