The move to cloud is generally doing away with the need for servers on premises. Cloud services fall into three areas IaaS, PaaS, SaaS. An often used analogy is how a pizza is made where managing the pizza yourself is the equivalent of having servers in house and being served pizza at a restaurant being SaaS. There are many examples of web services in these three options. Amazon Web Services is a good example of IaaS and Microsoft Azure as IaaS or PaaS.
Many customers have already reserved their business name as a domain using services like 1and1 or GoDaddy. Often the host agency includes email and a basic web site. We usually recommend SME's keep DNS and web site separate from Office 365/Azure.
Azure provides most of Microsoft’s many offerings including virtual servers with many operating environments and sizes, multiple storage options, databases, networking, remote desktops, and data analysis to name a few. This includes RedHat/Centos Linux. Generally customers wanting more complex Azure employee their own internal ICT support.
Office 365 sits on Azure and is regarded as a SaaS. When using Office 365 you can leverage Azure Secruity using Enterprise Mobility + Security (EMS) providing levels of cloud Active Directory user control, Intune for mobile device management (MDM) and application management (MAM), and Azure Information Protection (AIP) to control data handling and loss prevention.