Ive searching for this particular question for a while, and I dont seem to find exactly an answer to it.
Lets imagine I have an Arduino Uno, which allows to remove the Arduino 328 chip very easily.
If I send to the chip a program (for example Blink), can I remove the 328 chip from the Arduino Uno shield, put it into a breadboard and add a power supply for the arduino Voltage and Ground pins (I think its 5V).
My question is: as soon as the chip has energy again, will it start running the Blink program?
You also need the 16mhz crystal + loading caps (typically 20pf, but it depends on the crystal), and 0.1uf decoupling capacitors between Vcc and Gnd, and between AVcc and Gnd, located as close to those pins as possible. Vcc and AVcc should be connected. Vcc and AREF should not be connected (some guides incorrectly show this).
And a pullup on reset is good practice, but not necessary (unless you're programming it via serial and want autoreset, then the 10k pullup is required)
Edit: Oh - and terminology note. A "shield" refers to one of the add-on boards that plugs into the connectors on the Arduino Uno/Leo/Mega, not the Arduino board itself.
That tutorial is incomplete though. It omits the necessary 0.1uf decoupling capacitors, so the result won't work reliably (random resets and problems writing the flash are the normal consequences - all highly intermittent and would be maddening to debug if you weren't aware they you were supposed to have decoupling caps)
I don't know who made that tutorial, but it is really pathetic that it has been up like that, everyone knows it's wrong and routinely warns people about it - but the wrong information stays up.