We present a detailed analysis describing the utility of the formaldehyde (H2CO) molecule in the derivation of the kinetic temperature and spatial density within molecular clouds. Measurements of 13 transitions from both the ortho and para species of H2CO have been made toward a sample of 11 active star Formation regions. These H2CO transitions range in frequency from 21 1 to 365 GHz and in upper-state energy from 21 to 241 K. This range in excitation has allowed us to analyze H2CO sensitivity to both cool (T(K) less-than-or-similar 50 K) and warm (T(K) > 50 K) molecular material. Using a spherical large velocity gradient (LVG) model to solve for the excitation of H2CO, we analyze the sensitivity of several ortho- and para-H2CO transition intensity ratios to the kinetic temperature and spatial density within molecular clouds. Through this analysis we derive several ''rules of thumb'' which should be followed when measurements of a particular intensity ratio are used to calculate T(K) or n(H-2) in a molecular cloud. We find that for T(K) less than or similar 150 K and over ranges in H2CO Column density typical for most molecular clouds, several H2CO transition intensity ratios are excellent monitors of T(K) and n(H-2). Since the transitions whose relative intensities are sensitive to kinetic temperature can be measured using the same receiving system (and can in some cases be measured within the same spectrum), calibration uncertainties are minimized. We also present a detailed analysis of the uncertainties encountered in our modeling procedure, including the potential importance of infrared excitation. Using our measured H2CO radiation temperatures, we have constrained LVG model solutions for the kinetic temperature, spatial density, and H2CO Species column density in each of the sources in our sample. Our derived spatial densities are comparable to those estimated using other molecular tracers. In all of the regions in our sample, though, we measure kinetic temperatures greater than 50 K, significantly higher than previous estimates for many of these sources. Previous underestimation of T(K) is due to the use of tracers which are sensitive only to cool (T(K) less than or similar 50 K) gas. In particular, temperatures of 100 K or more occur both toward the cool young binary object IRAS 16293-2422 and toward the ''protostellar condensations'' FIR 4/FIR 5 in NGC 2024. Using our ortho- and para-H2CO measurements we have calculated the N(ortho-H2CO)/N(para-H2CO) ratio in several of the sources in our sample. Our measurements indicate that N(ortho-H2CO)/N(para-H2CO) < 3 for most of our sources. When combined with the relatively high kinetic temperature in these objects, this N(ortho-H2CO)/N(para-H2CO) ratio suggests that dust grains might play an active role in H2CO chemistry.