The voltage ratio matches the turns ratio

1. The equation to recall is $\frac{V_p}{V_s}=\frac{N_p}{N_s}$, where $p$ stands for the primary coil and $s$ for the secondary.
2. $V$ is the voltage across each coil and $N$ is the number of turns wound on it.
3. Because both sides are ratios the units cancel, so volts or kilovolts both work, as long as both voltages are in the same one.
4. More turns on the secondary gives a larger secondary voltage, which is the step-up case.
5. Fewer turns on the secondary gives a smaller secondary voltage, which is the step-down case.
6. Only the ratio matters, not the actual numbers, so $100$ turns to $50$ turns behaves exactly like $2000$ turns to $1000$ turns.

For an ideal transformer, the power out equals the power in

1. A transformer that wastes no energy is 100% efficient, and then the power delivered by the secondary equals the power supplied to the primary.
2. Since $P=IV$ for each coil, that gives the equation $I_p{V}_p=I_s{V}_s$.
3. Stepping the voltage up therefore steps the current down by the same factor, and the other way round.
4. A transformer cannot create energy; all it does is trade voltage against current.
5. That is why the secondary of a step-down transformer is wound with thicker wire: it has to carry a much larger current.
6. Real transformers are not quite perfect, so a little energy is always wasted heating the coils and the core, and the output power is slightly less than the input.