Zotero allows you to create "Collections" and "Subcollections" in order to organize your citations.
When you first use Zotero every citation you save goes to the same place: My Library. And, in fact, every citation you ever save will go into this collection by default. But you can create your own "Collections" under My Library. A single citation can be assigned to any number of collections. (Because of this, it is best to think of Collections as Views, rather than as Folders).
You can create a collection by clicking on the New Collection icon: 
. Name your collection, and by default it will appear under the My Library collection.
You can always see the cites saved to a specific collection by clicking once on the collection. The main Zotero panel will then show all the citations in that collection, and that collection will be highlighted.
When you save a new citation from the Web to Zotero, it is always saved to two places:
1) The currently highlighted collection
2) My LIbrary
You can create a subcollection by right-clicking on a collection and choosing "New Subcollection" from the pop up menu.
Collections and subcollections appear in alphabetical order, but they can be moved into our out of a parent collection by dragging and dropping.
A very common error is to think that you are saving citations to one collection, only to discover that somehow you had highlighted a different collection before saving the citations. Fortunately, you can move citations from one (sub)collection to another by dragging and dropping.
When you are looking at all your items in My LIbrary, it can be useful to see what collection(s) an item is in -- this is especially true if you are looking at search results. With an item highlighted in the center panel, hold the ctrl key down and the collection that that item is in will be highlighted in the left hand panel.
Remove a citation from a specific collection by right clicking the citation while viewing the collection. Choose "Remove Item from Collection" from the pop up menu. This will not remove your citation from other collections: At the very least, you will still have one copy of this citation in your My Library collection.
Delete a citation completely by right clicking on the citation from any collection, and choosing "Move Item to Trash".
Filter citations within a collection by searching by keyword in the searchbox just above the main panel. Note that you are only searching the currently highlighted collection. If you want to search all your citations, make sure My Library is the current collection.
Sort citations by field using the field headers in the main panel.
Add additional fields to the default view by clicking on the
icon. Once you have added a field to the default view, you can sort by that field.
Dedupe by looking in your left hand panel, underneath your collections, and choosing "Duplicate Items". This will display all your apparent duplicates next to each other. You can review all duplicate copies, and delete those you don't need.
Find citations that are not assigned to any collection (other than My Library) by selecting "Unfiled Items" in the left hand panel below your collections. It is ok to have citations that are unfiled, but you may want to review these citations occasionally to see if they'd be better off in an existing or new (sub)collection.
An item that has an associated child item will have an arrow just to the left. When the arrow is putting to the right, the child items are hidden. When it is pointing down, you will see the child items listed underneath the parent. You can change the arrow direction by cliicking on it.
Parent items are usually citations. Child items are usually full text PDFs, full text snapshots, or notes that you have written. When you move a parent item, child items go with it.
Here is what it looks like in your Zotero main panel when your parent items are closed. The Journal Article and the Web Page below both have right-facting arrows, indicating child items. The Book does not (which is typical for books):
And here is what it looks like after you have clicked on the right-facing arrows. The arrows are now down-facing, and the child items are visible. In this case, the full text PDF of the journal article that came from a scholarly database, and a full-text snapshot of a webpage that came from a site that Zotero couldn't recognize :
