Storage and networking servicesYou'll find that most providers now support both object and block storage. Object storage is a distributed system for large data such as multimedia or email archives or VM images, where high latency is not a big deal. Block storage typically costs more and suits applications that are more performance-sensitive.
A cloud storage gateway is an appliance that sits in your own data center. Rather than back up data to local storage, it goes to the IaaS provider's data center through the cloud storage gateway, which might deduplicate or encrypt the data in the bargain. Historically, this has been one of the most popular uses of the cloud.
As for networking, the so-called virtual private cloud is all the rage now. An IaaS provider that supports the virtual private cloud enables you to set up multiple subnets with appropriate security tiers and select your own IP address ranges. Additionally, you'll want the ability to create a dedicated VLAN that directly connects your data center with the virtual private cloud.
Finally, most Web applications, particularly those that that involve rich media, will benefit from a CDN (content delivery network), which dials down user frustration -- globally, depending on the number of edge servers on the network. Here again, some providers have their own vast CDNs, and some partner with the likes of Akamai, but the bottom line is the ease of setup and the actual reduction in latency.
Database and app dev servicesYou can upload almost any database software you like to the cloud -- but it's much better to have database services maintained by the IaaS provider and ready to deploy. SQL database services are common, with MySOL the most prevalent, although you can also find enterprise-grade solutions with Oracle Database and/or PostgresSQL functionality. NoSQL databases (particularly MongoDB) are increasingly popular in the cloud, but at this stage you'll find some providers relying on partners to deliver the goods.