When you register for a Zotero account, you will be able to sync your library between computers and applications:

Go to www.zotero.org/user/register.
Provide a username, email address, & password.
Follow the instructions provided in the resulting confirmation e-mail to activate your account.

Here is a quick video on creating your account:

After registering your account, you will want to sync your library with the desktop application you just downloaded.

Screen shot of the Sync tab on Zotero preferences with the username and password filled in and the Sync automatically box checked.