A cloud server is a computer that provides cloud services to the client device. This usually comes in the form of processing services or storage services.
An example of processing services would be something like Opera Browser. When you put the Opera Browser into turbo mode, it will send processing request to the Opera server, so if you try to watch a video on your phone the Opera computer (cloud server) will process the video into smaller chunks and then send it to your phone (client). Another example would be Siri on the iPhone 4S. The iPhone doesn't really have voice recognition on it. When you speak to your phone the phone will record your voice and send it to one of Apple's computers (cloud server) and then Apple's computer will process your voice request and send back the answer to your phone (client).
Then there is cloud storage. An example of this is Picasa or Flickr. These photo storage companies are cloud servers and they hold pictures for you that you can then view on your phone or computer (clients). I have written two hubs about cloud storage on mobile phones. Click on my profile and check them out if you have time.