Amplifier analog design - how to effectively do it when gain varies so much?
If one adds negative feedback to an amplifier in such a fashion that an infinite-gain amplifier would have a gain of \$x\$, but the actual gain of the amplifier is not infinite but is instead \$y\$, the gain of the system is likely to be on the order of:

$$\dfrac{1}{1/x+1/y}$$

You can see that as \$y\$ approaches, the \$1/y\$ term drops out and the gain is simply \$x\$.  If \$y\$ is large relative to \$x\$, the net gain won't be quite equal to \$x\$, but even larges variations in \$y\$ will cause only small variations in the net gain.

If one constructs an amplifier with a suitably-biased input signal feeding the base of an NPN transistor whose emitter is tied to ground via resistor and whose collector is tied to the positive rail via another resistor, then the "ideal transistor" gain would be equal to the ratio of those two resistors, since the voltage on the emitter would track the voltage on the base, and the current through the collector would match the current through the emitter.  If the resistors were e.g. 1K and 10K, then increasing the base voltage by a 100mV would cause an extra 100uA to flow through the 1K resistor, which would in turn cause an extra 100uA to flow through the 10K resistor, thus causing it to drop an extra volt.

In practice, real transistors don't behave quite like ideal transistors, but if the resistors are chosen to yield a gain which is small relative to the transistor's beta, the effect of the transistor's beta on the output behavior will be relatively slight.

Op amps take this concept further, by using cascaded combinations of transistors so as to yield an inherent gain which is sufficiently high that variations in it become a non-factor.  The very high levels of gain often make it necessary for op amps to introduce some internal high-frequency negative feedback which will artificially attenuate gain so as to avoid unwanted oscillations; depending upon the particular components one is using, such limitations may be characterized tightly or loosely, but unless one is using an op amp near the limits of its performance they generally won't matter too much.