In EndNote Web, you can organize your citations into "groups". You can create as many groups as you want, and citations can go in more than one group. When you start with EndNote Web, you have one group "Unfiled", which is a catch-all group. In some cases, when you import citations from databases, those citations will go into Unfiled, so it's a good idea to check it regularly for new citations that you want to organize more specifically.
Organization should always be done in a way that makes sense to you. A common strategy is to make a group for each topic that you search for. Once you start writing your paper, make a separate group for the citations you actually use, so that you can easily find them. You might also make groups by assignment or by class. You can always change your mind, too.
Groups of citations can be shared with anyone who has an EndNote Web account.