Learn German: Ich lege das Buch auf den Boden. - I am putting the book on the floor.

auf is a two-way preposition. Because you are actively placing the book onto the floor, the action shows movement towards a destination (answering "where to?" or wohin?). Therefore, the masculine noun der Boden must take the accusative case, becoming den Boden. If the book were already resting there with no movement, you would use the dative case (auf dem Boden).

I always mix up legen and liegen. Why do we use lege here?

legen means "to lay" or "to put", which is an active movement. It takes a direct object (das Buch) and an accusative destination (auf den Boden). On the other hand, liegen means "to lie" or "to be located" (a stationary position). If the action were already finished, you would use the dative case: Das Buch liegt auf dem Boden (The book is lying on the floor).

Is das Buch in the nominative or accusative case here?

It is in the accusative case because it is the direct object of the sentence (the thing you are putting). Because Buch is a neuter noun, its accusative word for "the" (das) looks exactly the same as its nominative form.

Could I use the preposition an instead of auf for "on the floor"?