At the end of the day, it's about where is the data hosted and managed. If it is not on-premise then it is in the cloud. There are a lot of variations of cloud services, but if it is in a datacenter managed by a 3rd party then it is a cloud service.

The definition of cloud has really splintered in the past few years. My definition varies from some of the industries pundits. IMHO to be cloud the application or function must be delivered with the following characteristics: On-demand; Shared Resources, Internet Based and Consumption-based pricing. IaaS, PaaS and SaaS are all "subsets" of cloud. At it's most basic, is IaaS, in a sense, what a company has done is outsourced the infrastructure layer and some management of the infrastructure. PaaS, you have also "bought into" a platform for application development and delivery, and with SaaS, you have outsourced it end to end.
I believe the primary benefits an organization receives from have cloud in its IT strategy are: agility, elasticity, value-add.
Agility - the cloud give IT and consequently the organization as a whole the ability to move much more quickly. Resources can be provisioned on demand. Applications can be spun up on demand (depending upon the complexity of the application, of course). Business can be more innovative because the price of "failure" is so much lower.
Elasticity - we all talk about scalability. The infrastructure needs to scale (i.e. get larger). What we never talk about is the opposite. The ability to shrink in lean times. The cloud gives you the ability to do both, quickly and easily. Need 100 servers on Cyber Monday? No problem. Only need 5 after the holidays, no problem.
Value-add - to me, this is the most significant benefit, and one that is often overlooked. By leveraging the cloud, IT professionals are freed up from the day-to-day management of "keeping the lights on" and can spend far more time learning and understanding the business in which they operate. Armed with this understanding, they can bring technology to bear to solve business issues or provide advanced levels of innovation.

I believe another dynamic of the question relates to who is asking the question. As technology practitioners we are constantly evolving the meaning of cloud and the various types. Those outside of IT many times use it as the generic "not on premises" term. This is the new art in the science of technology.

I would say - Cloud means in today's enterprise IT world "You relax sir, we'll manage". That's how it is being sold and bought. Fundamentally there are hardware and software that decides your IT. In case of PAAS if you host your servers somewhere else other than your own data center - you call it cloud. Now, you don't have to hire people for your data-center. Cloud people will manage this. All you have to calculate your cost. So far, Cloud is winning comparatively your own data center cost. The similar example goes for SAAS services. In good old IT days, people used to write their own application. But in SAAS - a vendor writes it, hosts it in their environment, can customize for you and all you have pay as per the usage. Well, for generic solutions like CRM that's not a bad deal at all.