Separation axioms are properties of topological space which, roughly speaking, say in what way two points, a point and a closed set, or two closed sets can be "separated". Most important are $T_0$-spaces, $T_1$-spaces, Hausdorff, regular, completely regular and normal spaces.
The most important separation axioms are $T_0$-spaces (Kolmogorov), $T_1$-spaces (Fréchet), $T_2$-spaces (Hausdorff), $T_{2\frac12}$-spaces (Urysohn), $T_3$-spaces (regular), $T_{3\frac12}$-spaces (completely regular) (Tychonoff) and $T_4$-spaces (normal) spaces.