Solar flares accelerate particles up to high energies (MeV and GeV scales for electrons and ions, respectively) through efficient acceleration processes that are not currently understood. Hard X-rays (HXRs) are the most direct diagnostic of flare-accelerated electrons. However, past and current solar HXR observers lack the necessary sensitivity and imaging dynamic range to make detailed studies of faint HXR sources in the solar corona (where particle acceleration is thought to occur); these limitations are mainly due to the indirect Fourier imaging techniques used by these observers. With greater sensitivity and dynamic range, electron acceleration sites could be systematically studied in detail. Both these capabilities can be advanced by the use of direct focusing optics. The recently own Focusing Optics X-ray Solar Imager (FOXSI) sounding rocket payload demonstrates the unique diagnostic power of focusing optics for observations of solar HXRs. FOXSI features grazing-incidence replicated nickel optics with similar to 5 arcsecond resolution and fine-pitch silicon strip detectors with a similar to 7.7 arcsecond strip pitch. FOXSI flew successfully on 2012 November 2, producing images and spectra of a microflare and performing a search for non-thermal emission (4-15 keV) from nanoflares occurring outside active regions in the quiet Sun. A future spacecraft version of FOXSI, featuring similar optics and detectors, could make detailed observations of HXRs from flare-accelerated electrons, identifying and characterizing particle acceleration sites and mapping out paths of energetic electrons as they leave these sites and propagate throughout the solar corona. This paper will describe the FOXSI instrument and present images from the first flight.