Dimensional analysis means to obtain results by analyzing the units in question, etc. DO NOT USE THIS TAG if your question is about dimensions as in degrees of freedom.
An example of dimensional analysis would be constructing the Planck time. The Planck time is to involve the Planck constant, the gravitational constant, and the speed of light. So, we let the Planck time be products of these constants raised to unknown exponents. Then by requiring that this Planck constant has units of time, we calculate these unknown exponents.
Other instances of the usefulness of dimensional analysis include:
- Calculating other Planck units; e.g. Planck length, Planck mass, etc.