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The general word consultant is very vague, but mostly means Management Consulting. The basic idea is that you help companies fix problems that they cannot do on their own, for either time, resource (people), or technical knowledge constraints.
The types of careers you can have are, generally, strategy, operations, information technology, human resources, financial, engineering, etc. There are many types, but it comes down to what you want to do. The top tier consultancies are centered around Strategy and help companies make decisions that affect the whole organization, such as telling Pepsi to abandon soda and target energy drinks - a major, major move.