Summary is the most catch-all term of this group, and the one that shows up the most in general everyday English.
Abstract is most commonly used in the scientific context. It is typically a formal requirement for publication, as the initial section of a scientific paper. Often times if you find scientific papers online, it is just the abstract that is available.
Overview is similar in literal meaning to "summary". It has a slight informality to it.
Synopsis again could be exchanged directly for "summary" in most contexts. It has a slightly more formal feel, and shows up in the literature and the arts a bit more frequently than other contexts (e.g., "I just want to read a synopsis of the novel, not the whole thing" sounds a bit better than "summary"). A synopsis often is often more detailed than a regular "summary".
Executive Summary shows up most often in a business context, or sometimes also in a political context (e.g., think-tank white papers).
Any of these would probably work in a research report, but it would also depend on the audience. Scientists would probably be most comfortable with "Abstract"; MBAs with "Executive Summary"; for a more general public audience where you want to seem accessible, "Overview". If you're not sure, I can't imagine going wrong with "Summary".
Note that I'm American, so this answer applies most directly to American English.