A BioSketch is a very specific document that the NIH requires for almost all applications/awards/or progress reports. However, it is not confined to usage by the NIH. For example, the AHA required a BioSketch as part of the pre-doctoral fellowship application. So I think it is a fairly important document to know about. The NIH requirements can be found on their website. There are 2 format types: non-fellowship and fellowship. Depending on your situation, it should be pretty obvious which template you need to use, but if not, you can typically find out more in the details of your particular grant/fellowship. My main advice on this document, is to be aware of the dates at the top of the biosketch template provided by the NIH. In parentheses, there will be a date following the words 'Approved Through'. These biosketches have been around for a long time and the formating has varied through the years. So it is commonplace for a friend or labmate to send you their biosketch as a guide, but if their document is outdated and you reuse their template, you may submit a biosketch that is using an out-of-date template. It doesn't seem like a very big deal, but you don't want to give the institutions any reason to reject your application.
Administrative note: This biosketch requires an eRA commons user name. If you are a graduate student, then your PI will likely know what this is and you will probably already have an account. However, if you don't have an eRA commons account, it may take a few days to get set up and you don't want to run into this problem at the last minute.
Personal experience: This document requires you to list out your grades for all classes taken. If you have any dip in grades, as I did in my senior year, it is important that you at least address this in a sentence or two in your personal statement. I had a reviewer make a comment regarding my semester of poor grades and they wanted an explanation.
Personal statement: If you have some publications or national conference abstracts/talks, you can reference them in the personal statement section. Just make sure to not exceed the limit. When I was writing my application, the limit was 4 citations. You may also make referecnes in the 'Contribution to Science' section, but I wouldn't repeat any of the ctiations if you already have them in the personal statment section. Also it is good practice to put your name in bold in the references, so that the reviewer can easily find your name and position in the author list.
Other Experiences and Professional Memberships: You should probably be a member of whatever national organization represents your particular field. For example, I am in the biomedical engineering field and am a member of the Biomedical Engineering Society (BMES). If you are looking to beef this up, you can join organizations like the American Association for the Advancement of Science (AAAS) or look for student chapters of academic organizations at your university. SyBBURE could definitely be listed in this category.