Suppose that X is a topological space.
X is a regular space if and only if, given any closed set F and any point x that does not belong to F, there exists a neighbourhood U of x and a neighbourhood V of F that are disjoint. In fancier terms, this condition says that x and F can be separated by neighbourhoods.
X is a T3 space or regular Hausdorff space if and only if it is both regular and Hausdorff.
Note that some mathematical literature uses different definitions for the terms "regular" and "T3". The definitions that we have given here are the ones usually used today; however, some authors switch the meanings of the two terms, or use both terms synonymously for only one condition. In this encyclopedia, we will use the term "regular" freely, but we will usually say "regular Hausdorff" instead of the less clear "T3". In other literature, one should take care to find out which definitions the author is using. (The phrase "regular Hausdorff", however, is unambiguous.) For more on this issue, see History of the separation axioms.
Speaking more theoretically, the conditions of regularity and T3-ness are related by Kolmogorov quotients. A space is regular if and only if its Kolmogorov quotient is T3; and, as mentioned, a space is T3 if and only if it's both regular and T0. Thus a regular space encountered in practice can usually be assumed to be T3, by replacing the space with its Kolmogorov quotient.
There are many results for topological spaces that hold for both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand, those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.
There are many situations where another condition of topological spaces (such as normality, paracompactness, or local compactness) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied. Such conditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view, regularity is not really the issue here, and we could impose a weaker condition instead to get the same result. However, definitions are usually still phrased in terms of regularity, since this condition is more well known than any weaker one.
Most topological spaces studied in mathematical analysis are regular; in fact, they are usually completely regular, which is a stronger condition. Regular spaces should also be contrasted with normal spaces.
Thus, regular spaces are generally not studied because interesting spaces in mathematics are regular without also satisfying some stronger condition. Instead, they are studied to find properties and theorems, such as the ones below, that are actually applied to completely regular spaces, typically in analysis.
There exists Hausdorff spaces that are not regular. An example is the set R with the topology generated by sets of the form U - C, where U is an open set in the usual sense, and C is any countable subset of U.
Taking the interiors of these closed neighbourhoods, we see that the regular open sets form a base for the open sets of the regular space X. This property is actually weaker than regularity; a topological space whose regular open sets form a base is semiregular.