You recently purchased a new battery for your Dell Inspiron 1501. Before an old battery died, you got this message before a Windows logo came up. "The AC adapter type cannot be determined. This will prevent optimal system performance.
Please check AC adapter connect properly, remove AC adapter and plug it in again". You just started getting that message with your new battery. Is something else wrong?
There is a known issue with some of Dell adapters when a little metal pin gets bent or broken and a computer can no longer identify that adapter. If you can verify that that pin is bent or broken that would cause that error message and as long as it charges your battery and works then you can ignore that message. If you can't verify a bent or broken pin it might be some Dell software or a faulty AC Adapter.
You can remove Dell Quickset software and see if after rebooting your computer that message disappears. It could be that software indicator is wrong or a bent or broken pin on your AC adapter. You could test it with a Volt meter or take it into a computer shop to test it.