In the first part of this thesis, we investigate the entanglement entropy for the generalized charged BTZ black hole through the correspondence. The generalized charged BTZ solutions are derived by considering three types of nonlinear electromagnetic fields ( NLED ) coupled with Einstein gravity. Using the holographic description of the entanglement entropy for the strip-subsystem in boundary , we will find the first law-like relation between the variation of holographic entanglement entropy and the variation of energy of the subsystem in terms of the mass and the electric charge up to the second-order. Since the bulk Einstein quations and the equation of motion for gauge fields hold for all orders of the small perturbation, we expect that the first law-like relation to be also satisfied for the second-order in which the entanglement temperature gets modified. We also obtain appropriate counterterms to renormalize the energy tensor associated with the bulk on-shell actions. In the second part of this thesis, We study analytically the properties of the Weyl holographic superconductor in the Lifshitz black hole background, assuming that the back reaction effects are negligible and take the probe limit . We find that the critical temperature of the Weyl superconductor decreases with increasing Lifshitz dynamical exponent , z , indicating that condensation becomes difficult . In addition , it is found that the critical temperature and condensation operator could be affected by applying the Weyl coupling , . T he constraints imposed on the Weyl coupling in the Lifshitz background and in d dimensions will be initially explored by demanding that the dual CFT should respect the causality and that the energy flux be positive in all directions for the boundary CFT . Moreover , we compute the critical magnetic field and investigate its dependence on the parameters and z. Finally , we show numerically that the Weyl coupling parameter and the Lifshitz dynamical exponent z together control the size and strength of the conductivity peak and the ratio of gap frequency over critical temperature . In the last part of this thsis, we study phase transition of a two dimensional holographic superconductor with the holographic view of complexity . The case of 2D holographic superconductivity of our interest is based on the correspondence. We also consider the in f uence of the backreaction on the dynamics of perturbation in the background spacetime . Then we use the domain wall method to calculate the holographic complexity . Our result shows that this quantity is singular at critical chemical potential . Its means that singularities of the complexity happen at normal/superconductor phase transition points for 2D holographic super