Unsteady fluid systems are nonlinear high-dimensional dynamical systems that may exhibit multiple complex phenomena in both time and space. Reduced Order Modeling (ROM) of fluid flows has been an active research topic in the recent decade with the primary goal to decompose complex flows into a set of features most important for future state prediction and control, typically using a dimensionality reduction technique. In this work, a novel data-driven technique based on the power of deep neural networks for ROM of the unsteady fluid flows is introduced. An autoencoder network is used for nonlinear dimension reduction and feature extraction as an alternative for singular value decomposition (SVD). Then, the extracted features are used as an input for a long short-term memory (LSTM) network to predict the velocity field at future time instances. The proposed autoencoder-LSTM method is compared with non-intrusive reduced order models based on dynamic mode decomposition (DMD) and proper orthogonal decomposition. Moreover, an autoencoder-DMD algorithm is introduced for ROM, which uses the autoencoder network for dimensionality reduction rather than SVD rank truncation. The results show that the autoencoder-LSTM method is considerably capable of predicting fluid flow evolution, where higher values for the coefficient of determination R² are obtained using autoencoder-LSTM compared to other models.