A transformer consists of 2 coils of wire, usually wrapped around the same iron core to “couple” them together. An AC current in one coil of the transformer (usually called the “primary”) induces a changing magnetic field affecting both coils, which induces an AC current in the other coil (”secondary”). The ratio of the # of turns on the primary ($N_p$) to the # of turns on the secondary ($N_s$) mainly determines the behavior of the transformer.
Transformers are commonly found in microwaves, where 120V wall voltage is boosted to 2kV to generate microwaves. They are also used to transmit power over the grid — high currents cause transmission lines to heat up, which is dangerous and lossy, so voltage from generators may be boosted to ~20kV (and the current proportionally decreased) to send across transmission lines, then stepped down to 120V AC for home outlet use. They are 98-99% efficient.
The ideal transformer model consists of a primary and secondary coil with turns ratio $N_P : N_S$, and a voltage source connected to the primary and a load $Z_L$ connected to the secondary. The dots indicate the direction the coils are wound; convention is that current enters the dot (into the coil) on the primary and leaves the dot (away from the coil) on the secondary. The turns ratio is sometimes denoted $n = N_p/N_s$.
Roughly, we can think of the strength of the induced magnetic field as proportional to the number of coils, and a stronger oscillating magnetic field induces a higher voltage across an inductor. Thus the primary/secondary voltages are proportional to the number of turns:
$$ \frac{V_P}{N_P} = \frac{V_S}{N_S} \Longleftrightarrow \frac{V_P}{V_S} = \frac{N_P}{N_S} = n $$
In an ideal transformer, power is perfectly conserved, $I_P V_P = I_S V_S$
Combining this with the previous equation, we see that current is inversely proportional to the number of turns: $I_P N_P = I_S N_S$
How do you analyze currents/voltages when there are RLC components on both the primary and secondary? The currents will no longer match what is expected from the ideal transformer model. Reflection is a technique that lets us simplify transformer circuits by “reflecting” components onto a single side of a single transformer.
A voltage source with voltage $V_S$ on the secondary will have equivalent primary voltage
$$ V_P = V_S\left(\frac{N_P}{N_S}\right) = nV_S $$
Similarly, a current source of $I_S$ on the secondary will have equivalent primary current
$$ I_P = I_S\left(\frac{N_S}{N_P}\right) = \frac{I_s}{n} $$
Combining the 2 above equations with $Z=\frac{V}{I}$, a component with impedance $Z_S$ on the secondary side will have an equivalent impedance on the primary side of
$$ Z_P = \frac{V_P}{I_P} = Z_S \left(\frac{N_P}{N_S}\right)^2 = \boxed{n^{2} Z_S} $$
Components already in the same loop of the transformer need not be reflected. Finally, after reflecting all components onto a single “side” of a single transformer, we can find the voltages and currents through that loop, and convert those accordingly using the turns ratios.