When we start looking at Windows Azure there are several key workloads that come to mind, the ones that are pretty much slam dunks for being a fit.
Workload 1 - On and Off
On and Off typically represents applications or web sites that are needed for a short period of time and later shut off. An example of this would be a marketing campaign that runs a web site for two months every year around Christmas because it is your peak sales time. The cost of acquiring the capacity for this heavy load period and then maintaining throughout the year or recommissioning can often be very high. Azure allows you to minimize both the upfront capital investment and the recurring monthly maintenance expenses. The reason for this is that you pay for what you use in Azure, so if you run it for these two months then decomission it at the end you stop paying for it until the next year when you redeploy. This not only helps reduce cost but potentially offers the business the ability to execute on more campaigns than they would typically due to overall cost.
Workload 2 - Scale Fast
Scale Fast is reflective of applications or sites that are introduced by the business without a good handle on what the adoption rate will be. Typically you would have to build out extra capacity at the beginning beyond the anticipated level to ensure the best customer experience. Using Azure you can start with a couple instances of the application and write code that will automatically deploy additional instances as required based on actual usage. This allows the available capacity to closely follow the real demand for the site, keeping costs down while reducing the burden of capacity planning.
Workload 3 - Predictable Bursting
Predictable Bursting is typically seen in companies that have a seasonal component to them, such as retail with Christmas driving a large quantity of their sales or finance where RRSP season can spike in activity. Azure can be used in this scenario as well to reduce investment in additional hardware and infrastructure required to support the load during these bursts. There are two ways people typically handle this type of capacity "augmentation", either through load transference where they host the application, or some components of the application, in Azure during the burst or as an application extension. Application extensions can be an interesting scenario with the application running on premise during normal load states or periods and when additional capacity is required they introduce additional capacity in Azure, never eliminating the on premise portion. This is sometimes referred to as a hybrid application.
Workload 4 - Unpredictable Bursting
Unpredictable Bursting is the same as Predictable Bursting above except in the sense the burst can come at any time. An example of this is if the government ran a campaign that gave tax breaks on purchasing new appliances appliance retailers may all of a sudden realize a significant load increase on their web sites. This is load can be handled in the same way as the Predictable Bursting scenario, except the identification of the burst scenario would typically be automated to ensure it could react and scale on demand.
The above workloads do not make up a comprehensive list of scenarios where Azure can be leveraged, but it does identify the key four workloads where the value of Azure can be quickly recognized.