If you're working on an unfamiliar computer, there's another easy option for taking your Zotero library with you...assuming the owner of the computer will let you download Zotero. This option will not work unless the machine has Zotero installed.
On your home computer, click the Zotero gear menu, choose Export Library, and save the file to a USB drive or cloud-based storage account. Then when you get onto another computer that has Zotero installed, just click the gear menu, choose Import, and find your Zotero file to import it.
When you're done, make sure to export any changes and import the new file onto your own computer.
If you're regularly using more than one computer in your research, Zotero's sync feature can keep your library up to date on all of them. Zotero can store a copy of your library on the Zotero.org server and check it for updates whenever you open your library on a different computer. All your computers must be running the same version of Zotero.
First, set up a (free, of course) Zotero.org user account. Then take the following steps:
Repeat this configuration on each of your computers. Any updates you make on one of your computers will be reflected on the others. This even works to synchronize your library among Windows, Mac and Linux computers.
For more details and help troubleshooting sync problems, check the Zotero site.
With the release of Zotero version 4.0, it is now possible to choose which directory Zotero uses when you attach links to files. If you use Dropbox (or another similar file-syncing service), you can now save your citations in Zotero, attach links to files in a Dropbox folder, and have both synchronize with every computer you use. By doing this, all of your citations and saved electronic documents will be easily available where you are working, as well as being backed up in the cloud and your hard drives. Here are instructions to set up the sync and link files:
1) Sign up for a Dropbox account. 2GB of storage is free. 100GB is available for $99/year. [For cheaper options providing large amounts of storage, Google Drive offers 5GB free and 25GB for $30/year and SugarSync offers 5GB free and 60GB for $75/year. If you go with Drive or Sugarsync, just substitute them for Dropbox in the file examples below.]
2) Download or upgrade to Zotero 4.0. I recommend downloading the Standalone version and then installing the appropriate browser extension for Chrome, Safari, or Firefox.
3) Create a folder within Dropbox to store all of your saved digital documents such as PDFs of articles. (In my example, that folder is labelled "Zotero.")
4) In Zotero, change the setting for the Base Directory to the Dropbox/Zotero folder. The path is Preferences/ Advanced/ Files and Folders. Here's what it should look like:
5) Move all your saved files and folders with digital articles and ebooks to the Dropbox/Zotero folder. You can keep whatever folder organizational structure you have already and just move everything into the Zotero folder.
6) In Zotero, right-click on a citation for which you have a saved digital file, then choose Add Attachment, then click Attach Link to File. Find the appropriate file in the Dropbox folder and attach it.
7) Once you've attached the link to the file, as long as the settings are correct and the same on each computer, Zotero and Dropbox should synchronize with whatever computers you use, and the file should be retrievable through Zotero.