The common idea
Through both configurations - inverting and non-inverting, the same negative feedback idea is implemented as follows:
The input Vin and (part of) the output voltage Vout are compared by subtraction and the result is applied to the (differential) op-amp input so that the difference to be zeroed.
It is only in the method of subtraction - in the inverting configuration, it is implemented in a parallel manner and in the non-inverting one in a serial manner. This leads to significant differences in the properties of the two circuit solutions.
Configuration. In the inverting amplifier, the two voltages are of opposite polarity (referenced to ground) and summed through a R1-R2 resistor network. The output (difference) voltage is also referenced to ground and therefore no differential input is required (the unused op-amp non-inverting input is grounded). So an inverting amplifier can be made by an amplifier with a single ended input but it has to be supplied by a voltage with an opposite (to the input voltage) polarity. This means that if the input voltage is bipolar, the op-amp has to be split supplied.
Gain. It is exactly equal to the ratio of the two resistances (R2/R1) because the two voltages being compared are separate.
Input resistance. Looked at from the other side, inside the loop the two voltages are of the same direction, therefore they are summed. As a result, the current increases and the input resistance decreases (Miller effect). In other words, the op-amp output voltage neutralizes the voltage drop across the resistor R2 and only R1 is left to operate; so the circuit part after R1 can be thought as a short connection.
Configuration. In the non-inverting amplifier, the two voltages are of the same polarity (referenced to ground) but are connected in the opposite direction in the loop (- +, + - or + -, - +) so they are subtracted. The difference voltage is "floating" so an amplifier with differential input is required... but it can be single-supplied.
Gain is one unit greater than the ratio of the two resistances (R2/R1 +1) because the output voltage includes the input voltage within itself. If we use the voltage drop across R2 as an output voltage, then the gain would be exactly R2/R1 as in the inverting amplifier but the load would be floating.
Input resistance. Since the two voltages are subtracted, the current enormously decreases and the input resistance increases ("bootstrapping" effect). Seen from the input source side, the circuit is an open circuit.
I have illustrated these explanations by CircuitLab simulations in my answer to a similar question.