Language

In Automata-theory -

Alphabet :- Finite set of symbols.

String :- Sequence of symbols.

Language :- Set of strings.

Operations on Strings :- 
1. (+) - union

2. (.) - concatenation  

3. (*) - Kleene star 

Ex :-  A = "aa" B = "bb" 
1. A + B = {"aa","bb"} 
2. A.B = "aabb"

3. A^* = ε ∪A∪A²…

Notice that A^* the set of all finite strings over A.

Finite permutation of strings and above mentioned operations creates Regular Expression.

Regular Language :- 

Def 1 :- If -
             i. ∃ λ : Σ →  M is a  monoid homomorphism where M is a finite monoid.
             ii. S ⊆ M
then { ω∈ Σ^* | λ(ω) ∈ S} is regular.

Def 2 :- 
Let, L ⊆ Σ^*and there is an equivalence relation ~ s.t uw ∈ L if and only if vw ∈ L for all w ∈ Σ*
The language L is regular ↔  Number of equivalence classes of ~ is finite.

A language L on alphabet Σ is Regular ↔ It is equivalent to a Regular Grammar
↔ It will have an Equivalent DFA ↔ It will have an Equivalent NFA.