There are several types of services that cloud service providers offer. At a high level, understanding a cloud service providers responsibilities when it comes to your infrastructure, applications and data is important when evaluating whether those providers can fulfill your requirements.

On Premises/Traditional
This model leaves out the cloud provider altogether. This model was prominent up to the widespread adoption of cloud services in general. Most companies have a combination / Hybrid model. This allows them to offload some services to the cloud and maintain some in house. The On Premises model refers to equipment owned, operated, maintained, and supported by the company.
Your on premises network infrastructure wiring, switches, routers, access points and firewall equipment falls into this category even if they are managed by a third party or have cloud management capabilities.
Co-Location or Data Center Providers
While not a cloud service provider per-se, an intermediate step between on-premises and cloud services is a situation where you still own your equipment but the environment is looked after for you. Things like physical security, heating and air conditioning, upstream network connectivity, power and backup generators.
IaaS – Infrastructure as a Service
This is the first step into the cloud space. In addition to operating out of a cloud (remote) data center managed by a vendor, you are also using their servers, switches, NAS/SAN and other equipment. The IaaS cloud service provider is responsible for providing you a stable and reliable platform that is configured to your specifications. The provider is not required to provide the operating system (some may do this). Business choose IaaS as a first step into the cloud. It provides you a purely software managed solution without the expense associated with server hardware, network gear, or power considerations.
PaaS – Platform as a Service
Platform as a Service (PaaS) vendors go to the next level and provide the operating system, runtime and middleware of your platform. This makes the application and data the only infrastructure parts you are required to maintain. VDI (Virtual Desktops) or Remote Desktops are a good example of this type of service.
SaaS – Software as a Service
SaaS is the ultimate cloud service where even the applications are managed for you. A good example of this is MS Office 365, email providers, Salesforce / SugarCRM, Skype or Zoom. These are applications hosted in the cloud for your use for a subscription fee (usually per-user account).
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