In my own words, what an enterprise architect actually does is a mix of business and technology design, problem solving, change, and implementation. They do this through research, consultation, investigation, design, and advice. They work with a range of roles across the organisation including operational staff and management; and senior management and governance bodies.
Wikipedia describes the role as follows "Enterprise architects are practitioners of enterprise architecture; an enterprise strategic management discipline that operates within organisations."
The rest of the Wikipedia article is very good too - read the first two paragraphs to for nice summary of the Enterprise Architecture role: https://en.wikipedia.org/wiki/Enterprise_architect