Everyone will tell you something slightly different when it comes to what you should put in your book. I think its a bit more simple than it seems. Here's my list of the types of songs you should put into your audition book.
Contemporary Musical Theatre
Contemporary Musical Theatre normally refers to anything after RENT. These are your Dear Evan Hansen songs, your Light in the Piazza songs, etc. You should have at least a ballad and an uptempo in this section.
Classic Musical Theatre(Golden Age)
Classic or Golden Age Musical Theatre refers to the period between 1940's and the 1970's. Great examples include Oklahoma, Annie Get your Gun, and Most Happy Fella. Once again, You need at least a ballad and an uptempo from this category.
This one gets a little bit more tricky. I like to go by decades in this category. You need a 50's rock and roll song, a 60's/70's song, and 80's song, and a contemprary pop song. If you rap, put a rap song in your book. If you can do country, put a country song in your book. Don't neglect this section. It's just as important as the musical theatre rep. I suggest picking songs that are already built for piano. This prevents the acompaniment from being clunky or awkward. Of course, dont forget to include uptempos and ballads for each decade.
This one is almost optional. It's nice to have at least one song in here, in case youre going in for something unusual, but it's not imparitive. A Wild Card song can be anything you want it to be. It can be a song that you take and morph into a completley different meaning, it can be a role you'll never play, it could even be a classical aria to have just in case. A character song is something very specific that wont work for everything. For exapmle, Screw Loose from Crybaby. These are good to have, but wont be used quite as often.
These are just my opinions, of course. I like to tell people to take everything you hear(including my opinions) with a grain of salt. Everyone is different. Find what works for you. If you have any questions, my inbox is always open!