I'll expand a bit more on DH Justin's reply and give you the high level overview here.
DreamObjects is essentially an unlimited hard drive on the internet. Since it's also shared with other users, it's segmented by buckets and restricted by access and secret keys. Anyone can access data you store there by giving setting public permissions on those particular objects.
Extending the metaphor further, we're just providing the hard drive right now. There are applications that work with DreamObjects since we speak the same language as Amazon S3 and OpenStack Swift. Unfortunately not all applications are compatible because they don't allow you to configure which provider they communicate with. Generally speaking, they all point to S3. We're working on this.
DreamCompute is to servers as DreamObjects is to hard drives; it is "cloud computing." I know that term is everywhere right now, but I mean it in the traditional definition. With DreamCompute you'll be able to start multiple servers of varying sizes and use for whatever you need.
Both DreamObjects and DreamCompute are examples of IaaS, or Infrastructure as a Service. These infrastructure services need to be accessed via an API (and that's actually what's happening when you're using our panel or an app like Cyberduck). Because of these APIs, anyone building an app or service can use DreamObjects for storage or DreamCompute for computing resources.
You asked for examples of how you'd use each one...
DreamObjects can be used by anyone needing a place to backup or store data (i.e. using an app like Cyberduck). It can also be used by developers to store data for their own apps - think ElephantDrive or Dropbox using DreamObjects for storing your data.
DreamCompute is great for a web application. An application may have a couple database servers, a few application servers It's also useful for groups needing on-demand computing power - think payroll/billing systems that need to process data once a month or research projects crunching giat datasets. That's just a couple examples, but there are a lot more.