In the first part of this thesis we study the algebraic structure of integral domains and prove some basic properties of them. We give the examples of integral domains with the emphasis on sets of the form \(\mathbb{Z} + \mathbb{Z} \sqrt{m}\) and \(\mathbb{Z} + \mathbb{Z} (\frac{1 + \sqrt{m}}{2})\), where \(m\) is a squarefree integer. Moreover, associates, irreducible and prime elements of an integral domain with their properties are introduced. Also, for a deeper investigation of divisibility, we limit our discussion to a special class of integral domains - a principal ideal domain. In the second part, we discuss Euclidean domains especially among the sets \(\mathbb{Z} + \mathbb{Z} \sqrt{m}\) and \(\mathbb{Z} + \mathbb{Z} (\frac{1 + \sqrt{m}}{2})\). It is interesting that these sets are Euclidean domains with respect to the function \(\phi_m(r+s \sqrt{m}) = |r^2-ms^2|\) if and only if \(m \in \{-1, -2, -3, -7, -11 \}\) za \(m<0\) i \(m \in \{ 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 \}\) for \(m<0\) and \(m \in \{ 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 \}\) for \(m>0\). We apply these results to determine when an odd prime can be represented by a binary quadratic form \((x^2 + y^2, x^2 + 2y^2, x^2 + xy + y^2, x^2 + xy + 2y^2\) and \(x^2 + xy + 3y^2)\).