The DNA motif discovery is a primary step in many systems for studying gene function. Motif discovery plays a vital role in identification of Transcription Factor Binding Sites (TFBSs) that help in learning the mechanisms for regulation of gene expression. Over the past decades, different algorithms were used to design fast and accurate motif discovery tools. These algorithms are generally classified into consensus or probabilistic approaches that many of them are time-consuming and easily trapped in a local optimum. Nature-inspired algorithms and many of combinatorial algorithms are recently proposed to overcome these problems. This paper presents a general classification of motif discovery algorithms with new sub-categories that facilitate building a successful motif discovery algorithm. It also presents a summary of comparison between them.