Depending on the mobile device, you can check email using the mobile web client, IMAP, POP, or ActiveSync.
ActiveSync isn't available in the FOSS edition, or (apparently) the Basic (10 user) Zimbra appliance. There are 3rd-party add-ons that might provide this functionality, see Funambol and Z-Push.
Blackberry is a special case. You can use IMAP/POP, or you can install a 3rd-party ActiveSync client, or you can use the Blackberry connector in conjunction with a Blackberry Enterprise Server.
Of course from a user standpoint all of these options depend on what your Zimbra provider or administrator provides. You should consult them to see what's available.
See also
Mobile collaboration tools for enterprise messaging for mobile devices