I have a color sensor that senses color and converts to sound output using MP3 player.
The whole system can work when I use USB to power the Arduino nano.
But it stops working when I use battery.
Does anybody know why?
Post a schematic / picture of how things are connected, and links to the components.
Yes
Schematic.pdf (1.9 MB)
What battery?
Applied to what, where?
PE --
Reluctantly opened your PDF.
Why is there a 100Ω resistor in series with the speaker? (There ought not.) It likes an 8Ω speaker.
PP3 9V battery is totally unsuitable for your project. (In fact there are very few projects where that type of battery would be suitable.)
Switch to 4x AA NiMH rechargeables and connect directly to the 5V rail. But take care never to use 4x AA non-rechargeable, because that would be 6V which would damage the nano and other components.
Why DFplayer is connected to SPI pins instead of RX TX?
Looks like your pushbutton will short the power supply via the 100 ohm resistor.
You should not have 5v and GND both going to the push button.
Your post was MOVED to its current location as it is more suitable.
Could you also take a few moments to Learn How To Use The Forum.
Other general help and troubleshooting advice can be found here.
It will help you get the best out of the forum in the future.
Always the first thing I think of is that you are attempting to use "Vin".
Please - never try and use "Vin". 
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