Spiking Neural Networks (SNNs) are gaining interest due to their event-driven processing which potentially consumes low power/energy computations in hardware platforms, while offering unsupervised learning capability due to the spike-timing-dependent plasticity (STDP) rule. However, state-of-the-art SNNs require a large memory footprint to achieve high accuracy, thereby making them difficult to be deployed on embedded systems, for instance on battery-powered mobile devices and IoT Edge nodes. Towards this, we propose FSpiNN, an optimization framework for obtaining memory- and energy-efficient SNNs for training and inference processing, with unsupervised learning capability while maintaining accuracy. It is achieved by (1) reducing the computational requirements of neuronal and STDP operations, (2) improving the accuracy of STDP-based learning, (3) compressing the SNN through a fixed-point quantization, and (4) incorporating the memory and energy requirements in the optimization process. FSpiNN reduces the computational requirements by reducing the number of neuronal operations, the STDP-based synaptic weight updates, and the STDP complexity. To improve the accuracy of learning, FSpiNN employs timestep-based synaptic weight updates, and adaptively determines the STDP potentiation factor and the effective inhibition strength. The experimental results show that, as compared to the state-of-the-art work, FSpiNN achieves 7.5x memory saving, and improves the energy-efficiency by 3.5x on average for training and by 1.8x on average for inference, across MNIST and Fashion MNIST datasets, with no accuracy loss for a network with 4900 excitatory neurons, thereby enabling energy-efficient SNNs for edge devices/embedded systems.