Compressed sensing (CS) theory has been recently applied in Magnetic Resonance Imaging (MRI) to accelerate the overall imaging process. In the CS implementation, various algorithms have been used to solve the nonlinear equation system for better image quality and reconstruction speed. However, there are no explicit criteria for an optimal CS algorithm selection in the practical MRI application. A systematic and comparative study of those commonly used algorithms is therefore essential for the implementation of CS in MRI. In this work, three typical algorithms, namely, the Gradient Projection For Sparse Reconstruction (GPSR) algorithm, Interior-point algorithm (l(1)_ls), and the Stagewise Orthogonal Matching Pursuit (StOMP) algorithm are compared and investigated in three different imaging scenarios, brain, angiogram and phantom imaging. The algorithms' performances are characterized in terms of image quality and reconstruction speed. The theoretical results show that the performance of the CS algorithms is case sensitive; overall, the StOMP algorithm offers the best solution in imaging quality, while the GPSR algorithm is the most efficient one among the three methods. In the next step, the algorithm performances and characteristics will be experimentally explored. It is hoped that this research will further support the applications of CS in MRI.