How does a battery recharge and also give output to a circuit? I'm wondering about smaller batteries like those for game consoles, headphones, etc.

 

Battery is a chemical process. By combining chemicals you can make electricity, and by adding electricity you can break back apart chemicals.
There are rechargeable batteries in AA and AAA size.
Many of the rechargeable batteries use different chemicals. Watch out for 1.25 Volt batteries. Some consoles will work at 1.5 volt but not at 1.2V. There are 1.5V rechargeable. You need to read the fine print.