A language S is defined to be regular if and only if there is a DFA M so that S = L(M).
For example,
{s | s ∈ {0,1}*, s is a binary number that is divisible by 3}is a regular language.
The key observation about a regular language L is that you must be able to distinguish between a member of L and a nonmember of L by scanning the string, keeping track of only a fixed amount of information, independent of the length of the string that you are scanning.
It is easy to come up with languages that appear to require a memory that grows as the length of the input grows. For example, consider
W = {anbn | n > 0}In order to be a member of W, a string must consist of some number of a's followed by the same number of b's.
= {ab, aabb, aaabbb, aaaabbbb, …}
The obvious way to decide whether a string is in W is to count the number of a's and count the number of b's, then compare the two counts.
(There is also the issue of whether the string does not have an a after a b, but that is easy to add in.)
The problem is that you cannot count the number of a's with a fixed amount of memory. The count could be arbitrarily large.