The problem of pricing an American option written on an underlying asset with constant price volatility has been studied extensively in literature. Real-world data, however, demonstrate that volatility is not constant, and stochastic volatility models are used to account for dynamic volatility changes. Option pricing methods that have been developed in literature for pricing under stochastic volatility focus mostly on European options. We consider the problem of pricing American options under stochastic volatility, which has had relatively much less attention from literature. First, we develop a transformation procedure to compute the optimal-exercise policy and option price and provide theoretical guarantees for convergence. Second, using this computational tool, we explore a variety of questions that seek insights into the dependence of option prices, exercise policies, and implied volatilities on the market price of volatility risk and correlation between the asset and stochastic volatility. The speed and accuracy of the procedure are compared against existing methods as well.