The easiest way, for example, to explain cloud computing is that the term “cloud” is really a metaphor for the Internet. Instead of information being stored on your own computer and you having to purchase the software to manage that information, you store information on a server and you don't even know where it is. The software is purchased or rented, in a sense, from the service provider. It's very popular, for example, with businesses. They don't have to invest in the software and they don't have to worry about securing the information. That raises privacy concerns, because the organization, the business in this case, no longer controls the information at its business site; it may be managed by a third party.
With respect to the reference to deep packet inspection, in the current communications environment, information is sent in what we call “packets”. It doesn't matter whether it's a telephone conversation or music, it's sent in individual packets and your conversation gets mixed up in various packets and then reassembled at the end.
For example, an ISP, or for that matter a law enforcement agency, has the ability to look at those packets as they go through a certain point. There are many uses for that, which are perfectly legitimate, to ensure that no one is trying to hack into that information as it's flowing past that point. However, the concern is that potentially people can inspect this information. They can look at it to hunt for key words, for example, which they may then want to use for marketing purposes. That's where the privacy concerns come in.
The reason it's called deep packet inspection is because it goes down to that level where you get a sense of what people are communicating about, the key words they're using.
I hope that helps explain, at least to a degree, those two concepts.