In this paper, we propose an automated computer platform for the purpose of classifying Electroencephalography (EEG) signals associated with left and right hand movements using a hybrid system that uses advanced feature extraction techniques and machine learning algorithms. It is known that EEG represents the brain activity by the electrical voltage fluctuations along the scalp, and Brain-Computer Interface (BCI) is a device that enables the use of the brain neural activity to communicate with others or to control machines, artificial limbs, or robots without direct physical movements. In our research work, we aspired to find the best feature extraction method that enables the differentiation between left and right executed fist movements through various classification algorithms. The EEG dataset used in this research was created and contributed to PhysioNet by the developers of the BCI2000 instrumentation system. Data was preprocessed using the EEGLAB MATLAB toolbox and artifacts removal was done using AAR. Data was epoched on the basis of Event-Related (De) Synchronization (ERD/ERS) and movement-related cortical potentials (MRCP) features. Mu/beta rhythms were isolated for the ERD/ERS analysis and delta rhythms were isolated for the MRCP analysis. The Independent Component Analysis (ICA) spatial filter was applied on related channels for noise reduction and isolation of both artifactually and neutrally generated EEG sources. The final feature vector included the ERD, ERS, and MRCP features in addition to the mean, power and energy of the activations of the resulting independent components of the epoched feature datasets. The datasets were inputted into two machine-learning algorithms: Neural Networks (NNs) and Support Vector Machines (SVMs). Intensive experiments were carried out and optimum classification performances of 89.8 and 97.1 were obtained using NN and SVM, respectively.