The discovery of the γ-ray pulsar PSR J1836+5925, powering the formerly unidentified EGRET source 3EG J1835+5918, was one of the early accomplishments of the Fermi Large Area Telescope (LAT). Sitting 25° off the Galactic plane, PSR J1836+5925 is a 173 ms pulsar with a characteristic age of 1.8 million years, a spindown luminosity of 1.1 × 1034 erg s–1, and a large off-peak (OP) emission component, making it quite unusual among the known γ-ray pulsar population. We present an analysis of one year of LAT data, including an updated timing solution, detailed spectral results, and a long-term light curve showing no indication of variability. No evidence for a surrounding pulsar wind nebula is seen and the spectral characteristics of the OP emission indicate it is likely magnetospheric. Analysis of recent XMM-Newton observations of the X-ray counterpart yields a detailed characterization of its spectrum, which, like Geminga, is consistent with that of a neutron star showing evidence for both magnetospheric and thermal emission.