the inverting amplifier
The inverting amplifier is the workhorse of op-amp circuits. The signal enters through a resistor into the minus input, which the op-amp holds as a virtual ground; a feedback resistor runs from the output back to that same node; and the plus input is tied to ground. The output is an inverted, scaled copy of the input.
Its gain is minus Rf over Rin. Here is the one-line derivation. The minus node is a virtual ground at 0 V, so the input current is Vin over Rin flowing into the node. No current enters the input (Rule 1), so all of it continues through the feedback resistor Rf, and the output must sit at 0 minus that current times Rf. The result is Vout equals minus (Rf over Rin) times Vin. For example, with Rin of 10 kilohm and Rf of 100 kilohm the gain is minus 10, so a 0.5 V input gives a minus 5 V output.
Why this is revolutionary: the gain is set by a ratio of two resistors, not by the messy, drifting, part-dependent op-amp. Resistors are cheap and precise, so the gain is precise and stable. The price you pay is input resistance: the source sees only Rin, so this circuit loads the source. Choose Rin large enough not to overload the source, but not so large that resistor noise and bias-current errors creep in.
Pick Rin of 1 kilohm and Rf of 47 kilohm. The gain is minus 47, so a small 20 mV input becomes a minus 0.94 V output, inverted and 47 times larger.
Gain depends only on the resistor ratio, never on the op-amp's wild internal gain.
Its input resistance is only Rin, so it loads the source, and the minus sign means a 180-degree phase flip. Cascade two stages to get a non-inverted result with precise gain.