In this thesis, we study the Takiff vertex algebra of type \(A_1^{(1)}\). In the first part of thesis, we construct a vacuum module \(V^{k,\overline{k}} (TA_1^{(1)})\) for the Takiff algebra of the affine Lie algebra of type \(A^{(1)}_1\), and then we endow \(V^{k,\overline{k}} (TA_1^{(1)})\) with the structure of the vertex algebra and the vertex operator algebra. Motivated by the one-to-one correspondence between the restricted modules for affine Lie algebras and modules for the universal affine vertex algebra, we prove that the restricted modules for the Takiff algebra of the affine Lie algebra \(\hat{\mathfrak{sl}}_2\) are exactly modules for the vertex algebra \(V^{k,\overline{k}} (TA_1^{(1)})\). In addition, we define generalized weight module, weight module and highest weght module as important example of restricted modules for the Takiff algebra of the affine Lie algebra of type \(A_1^(1)\). The central charge of vertex operator algebra \(V^{k,\overline{k}} (TA_1^{(1)})\) is level-independent. Vertex operator algebras associated to Heisenberg algebra have this property and all of them are isomorphic for nonzero levels. In this thesis, we prove that this is also true for vertex operator algebras \(V^{k,\overline{k}} (TA_1^{(1)})\)for nonzero complex numbers \(k,\overline{k}\). After construction, it is natural to examine simplicity of vertex algebra. The central part of the dissertation is dedicated to studying the irreducibility of the vacuum module \(V^k (TA_1^{(1)})\) from which we derive the conclusion about simplicity. The proof is based on the calculating roots of the Kac determinant of the corresponding bilinear form. For studying in detail some vertex algebra, it is important to know it’s Zhu algebra. So, we define generators and relations and identify the Zhu algebra \(A(V^k (TA_1^{(1)}))\). Finally, in the last part of thesis, we study realisation of \(W(2,2)\) algebra (or Galilean conformal algebra) in \(V^k (TA_1^{(1)})\).