In the usual way, for a given site, youe browser encrypts the data with the site's key. Only the website can decrypt it. Note that it's your browser that does the encryption.

If you want to compress data (not using full webpage) like this nokia browser or opera mini, you need to compress before encrypting, and decompress after decripting, and that doesn't work using the above method. So the browser compress the data, encrypts the data with nokias or opera's key, nokia decrypts it on his servers, decompress the data, encrypts with the target site's key, and the target site decrypts the data.

It's like you have half of your browser on your phone, and half in the cloud. As most cloud applications, it works best if you trust the cloud service provider.