You can keep your Zotero library synced to the Zotero website in order to access it from any computer. The first step is to create an account on the Zotero website. Then go to Edit → Settings → Sync. Fill in your user name and password and check Sync Automatically. Data syncing (references, notes, links, tags) is free and has no storage limit.
You can also sync your file attachments. Note that the free storage limit is 300 MB. If you need more storage, you can purchase additional cloud storage from Zotero.