The Palomar Transient Factory (PTF) Orion project is one of the experiments within the broader PTF survey, a systematic automated exploration of the sky for optical transients. Taking advantage of the wide (3°.5 × 2°.3) field of view available using the PTF camera installed at the Palomar 48 inch telescope, 40 nights were dedicated in 2009 December to 2010 January to perform continuous high-cadence differential photometry on a single field containing the young (7-10 Myr) 25 Ori association. Little is known empirically about the formation of planets at these young ages, and the primary motivation for the project is to search for planets around young stars in this region. The unique data set also provides for much ancillary science. In this first paper, we describe the survey and the data reduction pipeline, and present some initial results from an inspection of the most clearly varying stars relating to two of the ancillary science objectives: detection of eclipsing binaries and young stellar objects. We find 82 new eclipsing binary systems, 9 of which are good candidate 25 Ori or Orion OB1a association members. Of these, two are potential young W UMa type systems. We report on the possible low-mass (M-dwarf primary) eclipsing systems in the sample, which include six of the candidate young systems. Forty-five of the binary systems are close (mainly contact) systems, and one of these shows an orbital period among the shortest known for W UMa binaries, at 0.2156509 ± 0.0000071 days, with flat-bottomed primary eclipses, and a derived distance that appears consistent with membership in the general Orion association. One of the candidate young systems presents an unusual light curve, perhaps representing a semi-detached binary system with an inflated low-mass primary or a star with a warped disk, and may represent an additional young Orion member. Finally, we identify 14 probable new classical T-Tauri stars in our data, along with one previously known (CVSO 35) and one previously reported as a candidate weak-line T-Tauri star (SDSS J052700.12+010136.8).