This is a minimal circuit to get an ATMega328 running solo, without the Arduino board and USB stuff
If you leave out the 16MHz crystal and use the onboard 8MHz clock, you can bring down the power consumption even more, but at lower speed, obviously
The thing to keep in mind that the Arduino pins and the ATMega chip pins aren't the same. There's some mapping translation involved if you're using the Arduino IDE to program one
Here's the comparison of Arduino to ATMega328 pins. They are indeed different