The executive board (often abbreviated e-board) of a student group or student government organization is a small committee that does most of the work of running the organization. It usually has three to five members, and it can be considered loosely modeled off of the US corporate governance system, where (often, but not always) a board of directors is headed by an executive committee of individuals entrusted with the day-to-day management of the company.
The roles of President, Vice President, Treasurer, and Secretary are most common under this structure, but there are a host of other possibilities. Being on an executive board looks good on your resume.
Not all clubs follow this model, but it is required by most of the governing boards.