You can keep your Zotero library synced to the Zotero website, in order to access it on any computer you like. The first step is to create an account on the Zotero website. Then go to Edit → Preferences → Sync. Fill in your user name and password and check Sync Automatically. Data syncing (library items, notes, links, tags) is free and has no storage limit.
You can also sync your file attachements. There is a free storage of 300 MB.