the plus operator

The Kleene star says 'zero or more', but very often you want 'at least one', like requiring a name field to have at least one character. The plus operator is the convenient shorthand for that: 'one or more repetitions'. It saves you from writing the same pattern twice.

If R is a regular expression, R+ (written with a superscript plus, here just R followed by a plus sign) denotes every string formed by concatenating one or more strings from L(R). It is defined as exactly RR*: one mandatory copy of R, followed by zero or more further copies. So a+ denotes { a, aa, aaa, ... } and, unlike a*, it excludes the empty string ε unless ε itself happens to be in L(R). The whole difference between star and plus is just whether the empty case is included.

Be careful: the plus sign here is the same character used for union, so context decides the meaning. As a binary operator between two expressions, R+S means union; as a postfix operator after one expression, R+ means one-or-more. Most theory texts prefer to write union with + and use a superscript plus for one-or-more, while many programming regex engines write union with the vertical bar | and reserve + for one-or-more. The plus operator is a derived shorthand, not a fourth fundamental operation: anything you write with + you could rewrite using star and concatenation.