You are referring to DOMS: “Delayed Onset Muscle Soreness.”
“DOMS” is thought to be a result of tiny microscopic tearing of the muscle fibers. It also seems that the “eccentric” motion (lowering the weight) causes more soreness the next day than the “concentric” (lifting the weight) motion. By the way, the eccentric movement is also thought to be at least twice as important at producing greater strength and size gains! The old timers were instinctively aware of this that’s why they did negatives!
DOMS usually peaks between 24 and 72 hours after a training session. If it takes longer than that you really over did it!
When you are not feeling the soreness the next day that does not mean that you are not building muscle. I have to go out of my way at this point in my life in order to feel any soreness the next day. This means I must change up movements regularly and push a bit harder if I want to feel sore the next day (personally I like the feeling, for many reasons…but then that’s just me :).
Actually, as long as you are training regularly, training hard and eating properly, I would not worry about soreness the next day.