The Graduate Record Examination, aka the GRE, is just a standardized test used as part of most graduate school and graduate fellowship applications. Like the SAT and ACT, it is created and administered by a corporation, in this case, Educational Testing Service (ETS).
If you are considering applying for graduate school, particularly Ph.D. programs and associated fellowships, such as the Ford Foundation Pre-Doctoral Fellowship, then you will probably need to take the GRE before you apply (scores are valid for up to 5 years after you take the test). This post is specifically about the GRE General Test, but there are also specific subject tests which you may need to take depending on the field you are trying to apply to (See this page for more information on GRE Subject tests)
GRE test scores, though they are widely asked for, are rarely going to be a make or break factor in any application. Most things look at your application holistically and will not hyper focus on your score. That being said, a better score can also only help your chances. Further there is no definite breakdown of what a good or bad score looks like, especially given the range of things and fields people apply to with these scores (There is no minimum score for being admitted to a graduate program at Harvard). The only thing that will be able to give you some context for your scores is the percentiles. These show you what percentage of test takers got a score lower than yours. But ultimately that accounts for very little since the GRE, like all standardized tests, is only based on how well you can take the test, not anything about you as a person.