The accessibility of the Martian atmosphere to spacecraft provides an opportunity to study an ionosphere that differs from our own. Yet, despite the half century of measurements made at Mars, the current state of the neutral atmosphere and its embedded plasma (ionosphere) remains largely uncharacterized. In situ measurements of the neutral and ionized constituents versus height exist only from the two Viking Landers from the 1970s. Subsequent satellite and remote sensing data offer sparse global coverage of the ionosphere. Thermal characteristics of the plasma environment are not well understood. Patchy crustal magnetic fields interact with the Martian plasma in a way that has not been fully studied. Hence, investigating the coupled compositional, thermal and crustal-field-affected properties of the ionosphere can provide insight into comparative systems at Earth and other planets, as well as to atypical processes such as the solar wind interaction with topside ionospheric plasma and associated pathways to escape. Ionospheric models are fundamental tools that advance our understanding of complex plasma systems. A pre-existing one-dimensional model of the Martian ionosphere has been upgraded to include more comprehensive chemistry and transport physics. This new BU Mars Ionosphere Model has been used to study the composition, thermal structure and dynamics of the Martian ionosphere. Specifically: the sensitivity of the abundance of ions to neutral atmospheric composition has been quantified, diurnal patterns of ion and electron temperatures have been derived self-consistently using supra-thermal electron heating rates, and the behavior of ionospheric plasma in crustal field regions was simulated by constructing a two-dimensional ionospheric model. Results from these studies were compared with measurements and show that (1) ion composition at Mars is highly sensitive to the abundance of neutral molecular and atomic hydrogen, (2) lighter ions heat up more efficiently than heavier ones and provide additional heating sources for cooler plasma, and (3) crustal field morphology affects plasma dynamics and structure at Mars in a way that is consistent with observations. Finally, model predictions of ion composition and plasma temperatures are provided for observations to be made by several instruments on board the upcoming 2013 MAVEN orbiter.