Scale sets provide a way to create multiple VMs (based on the same image) dynamically as demand grows and shrinks.
Availability sets are a way of grouping VMs to spread them across multiple fault domains and update domains to provide high availability. Availability sets are a way to ensure that VMs are spread across different physical servers, switches, power supplies, etc, so that a failure of these does not affect all VMs.
For multi-tier applications with interdependent VMs be sure to put each tier in its own availability set to ensure that all tiers do not end up in the same fault or update domain.
Availability zones are separate data centers with their own power, cooling and networking.