\mathrm is the normal upright Roman font

\mathnormal is the normal math italic font: $\mathnormal{a}$ and $a$ give the same result

\mathcal is the special calligraphic font for uppercase letters only

\mathbf gives upright Roman boldface letters

\mathsf gives upright sans serif letters

\mathit gives text italic letters: $different\ne\mathit{different}$

\mathtt gives upright letters from the typewriter type font