What are room modes?
Standing waves: at frequencies whose half-wavelength fits a whole number of times between two parallel surfaces, the reflected sound reinforces itself. The result is bass that booms at some frequencies and positions in the room and cancels at others - the dominant acoustic problem in every small room.
How are room modes calculated?
Axial modes (the strongest kind, between one pair of surfaces) follow f = n × c ÷ 2L, where c is the speed of sound (343 m/s), L the distance between the surfaces, and n the mode order (1, 2, 3...). A 4 m room dimension has axial modes at about 43, 86, 129, 172 Hz. Tangential and oblique modes exist too but are progressively weaker.
What do I do about my room modes?
In order of leverage: position - move the listening spot and speakers so you sit away from pressure peaks (avoid room center and walls; ~38% of room length is a common start); bass trapping in corners, where all modes terminate; and EQ, which can tame a peak at one listening position but cannot fill a cancellation null. Verify what the calculator predicts with a slow sine sweep - your ears will find the modes immediately.
Which room dimensions are worst?
Cube-like rooms (equal dimensions) and rooms with dimensions in simple ratios (e.g. 3 m × 6 m) stack multiple modes on the same frequencies, doubling their damage. If dimensions in the table below produce modes within a couple of Hz of each other, those frequencies deserve the most treatment attention. Golden-ratio-ish rooms spread modes evenly - the reason studio designers obsess over dimension ratios.