We present a comprehensive photochemistry model for exploration of the chemical composition of terrestrial exoplanet atmospheres. The photochemistry model is designed from the ground up to have the capacity to treat all types of terrestrial planet atmospheres, ranging from oxidizing through reducing, which makes the code suitable for applications for the wide range of anticipated terrestrial exoplanet compositions. The one-dimensional chemical transport model treats up to 800 chemical reactions, photochemical processes, dry and wet deposition, surface emission, and thermal escape of O, H, C, N, and S bearing species, as well as formation and deposition of elemental sulfur and sulfuric acid aerosols. We validate the model by computing the atmospheric composition of current Earth and Mars and find agreement with observations of major trace gases in Earth's and Mars' atmospheres. We simulate several plausible atmospheric scenarios of terrestrial exoplanets and choose three benchmark cases for atmospheres from reducing to oxidizing. The most interesting finding is that atomic hydrogen is always a more abundant reactive radical than the hydroxyl radical in anoxic atmospheres. Whether atomic hydrogen is the most important removal path for a molecule of interest also depends on the relevant reaction rates. We also find that volcanic carbon compounds (i.e., CH_4_ and CO_2_) are chemically long-lived and tend to be well mixed in both reducing and oxidizing atmospheres, and their dry deposition velocities to the surface control the atmospheric oxidation states. Furthermore, we revisit whether photochemically produced oxygen can cause false positives for detecting oxygenic photosynthesis, and find that in 1 bar CO_2_-rich atmospheres oxygen and ozone may build up to levels that have conventionally been accepted as signatures of life, if there is no surface emission of reducing gases. The atmospheric scenarios presented in this paper can serve as the benchmark atmospheres for quickly assessing the lifetime of trace gases in reducing, weakly oxidizing, and highly oxidizing atmospheres on terrestrial exoplanets for the exploration of possible biosignature gases.