Impulsive particle acceleration and plasma heating at the Sun, from the largest solar eruptive events to the smallest flares, are related to fundamental processes throughout the Universe. While there have been significant advances in our understanding of impulsive energy release since the advent of RHESSI observations, there is a clear need for new X-ray observations that can capture the full range of emission in flares (e.g., faint coronal sources near bright chromospheric sources), follow the intricate evolution of energy release and changes in morphology, and search for the signatures of impulsive energy release in even the quiescent Sun. The FOXSI Small Explorer (SMEX) mission concept combines state-of-the-art grazing-incidence focusing optics with pixelated solid-state detectors to provide direct imaging of hard X-rays for the first time on a solar observatory. We present the science objectives of FOXSI and how its capabilities will address and resolve open questions regarding impulsive energy release at the Sun. These questions include: What are the time scales of the processes that accelerate electrons? How do flare-accelerated electrons escape into the heliosphere? What is the energy input of accelerated electrons into the chromosphere, and how is super-heated coronal plasma produced?