Tom Sawyer tells Huck that he can only join the band of robbers if he is respectable. This is ironic because robbers are not respectable.
Irony is when something is the opposite of what is expected. You would expect robbers to be liars and cheats, and outlaws. They are certainly not respectable. Tom and Huck have had their own experiences with outlaws, and know what they are really like. Yet Tom decides to create a band of robbers anyway.
But Tom Sawyer he hunted me up and said he was going to start a band of robbers, and I might join if I would go back to the widow and be respectable. (ch 1, p. 6)
Tom will only let Huck join the band of robbers if he is civilized, which means living with the widow and going to school. Of course this is completely ironic, because robbers are not civilized. This is a perfect example of Tom being out of touch with reality.
At this point in his life, Huck is trying to find himself. He admires and respects Tom Sawyer, so he follows his lead. Tom is playful and whimsical, and lives in a fantasy world that is completely the opposite of the harsh reality of the world that Huck lives in.
By beginning the book this way, Twain reminds us of the character of Tom Sawyer and uses him for ironic purposes. He is hinting to the reader that this book is going to be a bit more serious, but still funny.