A rate constant is an extremely useful quantitative characteristic of a chemical or physical process. The rate constant is the answer to the question, "What fraction of the molecules in the originating (or source) compartment is traveling via this process (or pathway) per unit time?" For example, if a rate constant is found to have the value 0.034 sec-1 (which is read as 0.034 per second), then 3.4% of the material in the source compartment travels by this pathway per second. Students often ask whether this means a rate constant must always be less than 1.0. The answer is, "No." A rate constant of 2.0 sec-1 means that if the mass in the source compartment remains constant (usually because of some balancing input from somewhere else) then twice that mass will leave the compartment by this pathway in the next second. If the mass is allowed to decline, the flow of material will decline accordingly. If the mass declines to zero, the flow of material will also be zero even though the rate constant remains at its finite value.
Another useful way to think about the meaning of a rate constant is to consider how it is related to the classical parameters of Henri-Michaelis-Menten enzyme kinetics. This is most easily seen in the linear range where the enzyme is not saturated. In this range, the flux of material through the pathway catalyzed by the enzyme is very nearly proportional to the substrate concentration, S. The constant of proportionality is determined by the ratio of Vmax to Km and is (apart from a volume factor) equal to the rate constant of the process catalyzed by the enzyme. Consequently, if you have been having trouble conceptualizing rate constants, try thinking of them as ratios of Vmax to Km. You will then see that anything which increases Vmax (viz. a change in the amount of the enzyme or in its turnover number) will increase the corresponding rate constant. Similarly, anything which increases the Michaelis constant of the enzyme (such as an allosteric modification) will decrease the corresponding rate constant.
Recognizing this relationship to classical enzyme kinetics should also make it easy to see why kineticists are fanatic about rate constants. When our analysis of the data reveals that a rate constant has changed in some experimental circumstance, we have the best possible evidence of a biologically significant regulatory change in the corresponding process. There are caveats here, but they belong in the advanced course.