The promise of applied Koopman operator theory is to find a linear representation of nonlinear dynamics. A closely related problem is that of finding a linearizing coordinate transform. The dynamic mode decomposition (DMD) is a data-driven approach to finding the best-fit linear operator which advances a set of measurements forward in time. However, DMD lacks a notion of observables, or functions of the measured state, which are an essential element of the Koopman operator's ability to linearly represent the evolution of nonlinear flows. We introduce the neural dynamic mode decomposition (NDMD), a model which takes advantage of the representational power of deep learning to discover an optimal set of observables for performing Koopman analysis via the dynamic mode decomposition. The NDMD model is trained on a multi-part loss function which includes terms designed to enforce (a) linear dynamics in the latent space of observables, (b) accurate, recurrently generated forecasts of future states, and (c) latent states which contain sufficient information to recover the original states. We demonstrate NDMD on damped and undamped nonlinear pendulum systems, as well as damped and undamped Duffing oscillator systems. NDMD learns a shared latent space where each trajectory evolves linearly. Each trajectory has its own associated time dynamics. To find a single approximation of the Koopman operator which generalizes to all trajectories, we reformulate NDMD to use training samples containing non-sequential snapshots from multiple trajectories. We modify the loss function to accommodate the non-sequential snapshots, and provide a procedure to use the fully trained model to produce a single, global approximation of the Koopman operator. NDMD is a data-driven decomposition method which can discover intrinsic dynamics of nonlinear systems. We extend NDMD to perform time-series classification tasks using the eigenvalues from DMD in the latent space as input features to a neural network classifier. This approach, which we call NDMD with classification, incorporates a classifier network and feature engineering steps into the end-to-end NDMD optimization problem. We conclude with a demonstration of NDMD with classification on an object discrimination task from acoustic recordings.