If you replaced the car battery with a fresh one and it's still dying, then something in the electrical system of the car is draining the battery while the vehicle is off. There are many things that could do that, such as an aftermarket alarm, aftermarket stereo, interior lights that are turned on or even something as silly as leaving your cell charger plugged in overnight. Electrical systems are a bit of a pain to diagnose so try looking at the obvious things first(lights, radio, alarm etc) and if that doesn't work have the car looked at by a qualified tech. Also have the alternator checked. There may be a possibility that the alternator is not charging your battery properly. A car will run on a fully charged battery for a while without you noticing anything during that day. The next morning the battery might be so drained that it doesn't have enough juice to turn over the starter and start the vehicle. Many shops will test the battery and charging system for very cheap if not free.
Hope that helps.