Mathematical models incorporating discrete time delays are utilized in scientific applications, because delays are intrinsic in the structure of many biological, physical, economic, and control systems. The following thesis is a study of a linear, scalar delay differential equation with two delays. The stability analysis given in this thesis uses the definitions, theorems and framework given by Mahaffy, Zak, and Joiner. After scaling the system has four model parameters: three coefficients and one delay term. For an interval of fixed delay values, the two- and three-dimensional regions of stability were located using the theory and definitions provided by Mahaffy, et al., in conjunction with the computational tools and routines developed during the research portion of this thesis. The regions of stability investigated displayed interesting geometric characteristics, many of which are summarized and illustrated in this work. One of those features are self-intersecting bifurcation surfaces that result in regions of stability that create "spur-like" formations on the main stability surface. Additionally, a complete characterization of the unbounded stability surface of the two-delay equation for a particular fixed delay is provided in this thesis. The evidence explained within also provides the basis for seeking a proof that the region of stability for that fixed delay is substantially larger than an already established guaranteed minimum.