Amazon charges for instances by the hours they're running and the type of instance. Think of an instance as a server, because that's what it is: an instance of a VM. You can find the prices for various services at http://aws.amazon.com/pricing/. What you want are EC2 pricing (for the VM instances) and EBS pricing (for the block storage for your disk volumes. For EC2 pricing figure out what size instances you need, then assume they'll be running 720 hours a month (30 days at 24 hours/day) and calculate the monthly cost. For EBS pricing take the number of gigabytes for each disk volume (each EC2 instance will need at least one volume for it's root filesystem) and multiply by the price (in dollars per gigabyte per month) to get your cost. You can manage instances the same way you would any other machine, other than usually needing to use SSH to get access and having to worry about firewalling (these are publicly-accessible machines, you can't shortcut on security by having them accessible only from within your own network).
The cost isn't actually too bad. For generic Linux, the largest general-purpose instance will, for a reserved instance on a 1-year commitment, cost you $987 up front and $59.04/month for runtime in the US West (Oregon) data center. An 8GB regular EBS volume will cost you $0.40/month for the space and $50/month for 1 billion IO requests. And not all instances need to be running all the time. You can, for instance, use on-demand instances for your testing systems and only start them when you're actually doing release testing, you'll need to pay for the EBS storage for their root volumes but you won't have any IO operations or run-time while the instance is stopped.
The downside, of course: if Amazon has an outage, you have an outage and you won't be able to do anything about it. This isn't as uncommon an occurrence as the sales guys would like you to believe. Your management has to accept this and agree that you guys aren't responsible for Amazon's outages or the first time an outage takes everything down it's going to be a horrible disaster for you. Note that some of the impact can be mitigated by having your servers hosted in different regions, but there's a cost impact from transferring data between regions. Availability zones... theoretically they let you mitigate problems, but it seems every time I hear of an AWS outage it's one where either the failure itself took out all the availability zones in the region or the outage was caused by a failure in the availability-zone failover process. This all isn't as major as it sounds, outages and failures happen running your own systems after all and you've dealt with that. It's more a matter of keeping your management in touch with the reality that, despite what the salescritters want everyone to believe, there is no magic AWS pixie dust that makes outages and failures just vanish into thin air.