Dimensional Analysis
Dimensional analysis is the method of tracking units through a calculation, canceling them like factors to check and convert results.
Definition
Dimensional analysis is a way of solving unit problems by writing the units next to the numbers and crossing them out in pairs, top and bottom, until only the unit you want is left. The factor-label method treats units as algebraic quantities that multiply, divide, and cancel, so a problem is solved by chaining conversion factors until the starting unit disappears and the arithmetic follows from the setup; it also checks formulas, since both sides of a valid equation must have the same units. More broadly it verifies that equations are dimensionally homogeneous, exposes missing constants, and can predict the form of a relationship from the variables involved. Arguments of functions such as $\sin$ and $\exp$ must be dimensionless.
Example
Seconds in $3$ minutes: $3\ \text{min} \times \frac{60\ \text{s}}{1\ \text{min}}$, and the "min" cancels to leave $180\ \text{s}$. A cart traveling $150\ \text{cm}$ in $1.8\ \text{s}$ has speed $\frac{150\ \text{cm}}{1.8\ \text{s}} \times \frac{1\ \text{m}}{100\ \text{cm}} = 0.83\ \text{m/s}$, and $\frac{13.5\ \text{g}}{5.0\ \text{cm}^3} = 2.7\ \text{g/cm}^3$ confirms by its units that density is mass over volume, not the reverse. A cart's speed from height $h$ can depend only on $h$ and $g$, and the only combination with dimensions of speed is $\sqrt{gh}$, since $\sqrt{\text{m/s}^2 \times \text{m}} = \text{m/s}$; the method cannot supply the $\sqrt{2}$ in $v = \sqrt{2gh}$, but it predicts that doubling $h$ multiplies $v$ by $\sqrt{2}$.
Key Insight
If the units do not cancel to what you want, the fraction was set up upside down; flip it and try again. Units are a free error-check on every calculation, and an answer in $\text{s/m}$ when you wanted speed tells you the fraction was inverted before you even look at the number. A dimensionally consistent equation can still be wrong, but a dimensionally inconsistent one is always wrong, so checking dimensions is the fastest way to reject a bad formula.