My dell laptop had one problem. I need battery and AC cord both on the laptop to turn on it. Either battery or AC cord did not on the laptop, I can not turn on laptop power. I don't know what is this problem / How to fix it ?

Many more recent laptops will not run on the AC adapter alone - the main battery must be installed too, and the battery must be charged enough such that it has at least a little capacity available. With older laptops you could often run them on the AC adapter alone. If your laptop won't run on the main battery alone, it's probably not charged enough for whatever reason - it's too old, or defective (e.g. internally shorted, a common problem), or, much less likely, the AC adapter or the circuits in the laptop to do with charging are not charging the battery properly, or the plug on the end of the AC adapter cord at the laptop end or the jack in the laptop is damaged. If the battery is more than two years old, the most likely thing is the battery needs to be replaced.

Also, check to see if the jack that the AC adapter plugs into is loose. It you can wiggle it with your fingers, it might have broken loose from the motherboard.