Windows, Mac, Linux. Pretty much any OS that has an upto date Web Browser, IMAP or POP3 email client or can interface with MS Exchange. See Desktop compatibility for more details.
For a complete list of supported operating systems for Zimbra Network Edition, see http://www.zimbra.com/products/downloads.html And for Zimbra Open Source Edition, see http://www.zimbra.com/community/downloads.html
No, all you need is a modern Web Browser: Fire Fox 3+, Microsoft Internet Explorer 7+, Apple Safari, Google Chrome are all supported.
Zimbra at this point in time do not officially support Centos. However, Agileware and many others in the Zimbra community have large production Zimbra deployments on Centos which are operating perfectly. As explained on the Centos website: "CentOS conforms fully with the upstream vendors redistribution policy and aims to be 100% binary compatible". There is a Request for Enhancement with Zimbra to official support Centos, see this Official Support for ZCS Network Edition on CentOS RFE #23487. For more details on this topic, see Zimbra CentOS support posts.