To determine where your Zotero data is being stored, go to the Advanced tab of Zotero Preferences and look under "Files and Folders." The default directory is listed under "Data Directory Location."
All of the item information, notes, and attachments imported into your Zotero collection are stored locally in this directory on your computer. The Zotero library data -- item information, notes, tags, etc. -- are included in the zotero.sqlite file. Your attachments -- PDFs, web snapshots, audiofiles, etc. -- are in the storage folder. If you would like to be able to access your collections from any computer connected to the Internet, then you must register with Zotero for an account and sync your data between your computer and Zotero's servers.
Everything in your collections, including notes, links, and tags, will be saved to the server. There is no limit to the amount of data that can be synced. (Note this is different from File Syncing, which is covered below.)
After you have registered with Zotero, open Zotero's Sync preferences tab and enter your Zotero account login information in the Data Syncing section.
At the next screen you will be provided with options for Data Syncing. If you choose Sync Automatically, each time changes are made, Zotero will automatically sync your data. If you disable this option, you can sync manually at any time by clicking the “Sync with Zotero Server” button () in the upper right corner of the Zotero application. Attached PDFs are only indexed on the computer where they are added. "Sync full-text content" syncs the index across computers, ensuring that the full text search includes all files, regardless of where they were indexed.
By default, Zotero will merge your local Zotero library with your library on zotero.org—any changes you make in one place will be applied to the other and on all other synced computers. If an item has changed in multiple places between syncs, you'll receive a conflict resolution dialog asking which version you'd like to keep. In the rare case that you want to completely overwrite your server library with the contents of your local library or vice versa, you can use the Sync Reset Options.
In the image above, the options for File Syncing with Zotero are selected. Data syncing syncs library items, but doesn't sync attached files (PDFs, audio and video files, images, etc.). The options here allow you to set up file syncing to accompany data syncing, using Zotero File Storage. Each Zotero user is given 300 MB of free Zotero File Storage for attached files, with larger storage plans available for purchase.
Another option for remote storage of files is to use a WebDAV service. Zotero does not provide support for third party WebDAV providers and when you set it up, you should make sure to use the "Verify Server" option to ensure that Zotero can successfully verify the WebDAV account.
If you use the ZotFile plugin for Zotero, you can use any cloud service to keep your library synced across computers, such as Dropbox or Sync. See the ZotFile tab for instructions.