Zotero allows you to easily create folders in order to organize your research citations.
Click the "New Collection..." button on the upper left side of your Zotero pane to create a new folder. Simply drag and drop your resources into the appropriate folder. This is a great way to organize your research sources according to particular assignments/projects that you're working on.
Your list of folders will be automatically alphabetized. If you'd like items to appear at the top of the list don't aren't alphabetically at the top, you right click on the folder and choose "Rename." This will allow you to add a special character such as an asterisk (*) to make sure the item appears at the top of your folder list.
If you have a PDF file that's arbitrarily named with the number from a given database (as seen above), right click on the PDF with your Zotero library and choose "Rename File from Parent Metadata." Zotero will take the associated metadata from the original resource and rename the PDF for you.
Since Zotero lives in the Firefox browser, using your Zotero login, you can access your saved citations and references from any computer on which Firefox and the Zotero plugin is installed.
Syncing works in two parts: data syncing and file syncing.
Data syncing transfers library citations, notes, links, tags, etc. (everything except attachment files) between the Zotero Firefox plugin and the Zotero servers, allowing you to work with your citations from any computer with the Zotero plugin and to view your library when you login at zotero.org.
While data syncing will sync library items, it will not transfer attached files: PDFs, audio, video, images, etc.
File syncing allows you to sync all of the above by using one of Zotero's two modes for syncing files: Zotero File Storage or use your own WebDAV account. You can also opt to save your Zotero library to a flash drive (just click on Advanced and choose the location you'd like your synced items to be saved).