What does "cup and Chaucer" mean?

I'll make an answer and kill the joke.

A man was reading The Canterbury Tales one Saturday morning, when his wife asked "What have you got there?" Replied he, "Just my cup and Chaucer."

The key to the pun in this case is "Saturday morning", where most cultures it is common practice to relax as a day off. Two common activities for relaxation (in general, and especially at this time and in combination) are reading and drinking a cup of coffee (or tea).

Taking the activities together, the pun is a play on the phrase "cup and saucer", where hot beverages are typically served on a saucer, which "is useful for protecting surfaces from possible damage due to the heat of a cup, and to catch overflow, splashes, and drips from the cup, thus protecting both table linen and the user sitting in a free-standing chair who holds both cup and saucer."

As JEL observes, in a different context the same play on words serves as a clever title for a café that also sells books.

So there you have it, a straightforward and thoroughly deceased play on words. You could also argue that this is an example of a dad joke, as evidenced by your reaction: you seem to have understood it was a pun, but it didn't make you laugh.