Brain-inspired cognitive computing has so far followed two major approaches - one uses multi-layered artificial neural networks (ANNs) to perform pattern-recognition-related tasks, whereas the other uses spiking neural networks (SNNs) to emulate biological neurons in an attempt to be as efficient and fault-tolerant as the brain. While there has been considerable progress in the former area due to a combination of effective training algorithms and acceleration platforms, the latter is still in its infancy due to the lack of both. SNNs have a distinct advantage over their ANN counterparts in that they are capable of operating in an event-driven manner, thus consuming very low power. Several recent efforts have proposed various SNN hardware design alternatives, however, these designs still incur considerable energy overheads. In this context, this paper proposes a comprehensive design spanning across the device, circuit, architecture and algorithm levels to build an ultra low-power architecture for SNN and ANN inference. For this, we use spintronics-based magnetic tunnel junction (MTJ) devices that have been shown to function as both neuro-synaptic crossbars as well as thresholding neurons and can operate at ultra low voltage and current levels. Using this MTJ-based neuron model and synaptic connections, we design a low power chip that has the flexibility to be deployed for inference of SNNs, ANNs as well as a combination of SNN-ANN hybrid networks - a distinct advantage compared to prior works. We demonstrate the competitive performance and energy efficiency of the SNNs as well as hybrid models on a suite of workloads. Our evaluations show that the proposed design, NEBULA, is up to 7.9× more energy efficient than a state-of-the-art design, ISAAC, in the ANN mode. In the SNN mode, our design is about 45× more energy-efficient than a contemporary SNN architecture, INXS. Power comparison between NEBULA ANN and SNN modes indicates that the latter is at least 6.25× more power-efficient for the observed benchmarks.