A cloud region is a geographic area, like AWS us-east-1 in Northern Virginia or Azure West Europe in the Netherlands. Inside that region the provider builds several isolated failure domains, and each one is an availability zone. AWS regions typically have three to six AZs. Each AZ is one or more buildings spaced a few kilometers apart from the others in the region.
The whole point of the separation is independence. Each AZ has its own utility power feed plus backup generators, its own cooling, and its own network links. A fire, a flood, a power substation failure, or a switch meltdown in one AZ is engineered to have no effect on the others. They are close enough that a fiber round trip between them stays in the single-digit millisecond range, but far enough that one physical disaster does not take out two of them.
An AZ is not the same as a region and not the same as a single rack or server. It sits in the middle: bigger than a rack, smaller than a region. When you launch a virtual machine, a managed database, or a disk volume, the provider places it in a specific AZ, and you usually get to choose which one or let the provider spread your resources for you.