I am replacing my tub's diverter spout, and I see that there are different connection types. How do I know if I need a slip-fit connection, an IPS connection, or a CxC slip-fit connection? What is the difference between these?

I am by no means an expert in this subject, but I thought I'd try and help you out anyway. I did some research for you, I hope this helps, if not I'm sorry :)

Here's the website : Tub Spouts and Diverters Product Info and Reviews | DoItYourself.com

Look under the spout next to the tile. If there's a small square hole then you have a slip on spout that's secured by a set screw located in the square hole These spouts are chrome over high impact plastic. Metal spouts, (chrome over brass ) have threads and screw on and off .I have put up images of the different types of spouts. (see images). Good luck, tom