I think the term cloud is abused by marketing but there are some aspects that I associate with the concept of the cloud.
A cloud environment is Infrastructure as a Service (IaaS) consisting of many services available on Demand on a pay-as-you-go basis.
Flexibility, automation, scalability and smooth interactions within the whole infrastructure are characteristics of the services.
Look at Amazon AWS. AWS is a public cloud service:
S3 - Scalable File Storage - flexible: pay by usage (used storage + traffic) without upfront cost; scalability: no limitations on storage + traffic (files size limit at 5TB); smooth interaction: CDN support through CloudFront, storage for all backup features of other AWS services
EC2 - Virtual machines - flexible: pay by usage, instant creation and destruction of unlimited numbers of machines; scalability: several different templates; smooth interaction: easy extension of virtual block storage devices + snapshots to S3
Route53 - DNS - flexible: pay-per-zone; scalability: unlimited (softlimit 100) in a geo-distributed virtual DNS-system; smooth interaction: API
and many more but you get my point.
The term private cloud means that you create a IaaS within your own datacenter and provide these services to yourself/your departments.
Keep in mind: Cloud services are always controlled by APIs because of the virtual and distributed nature of the infrastructure.