You need to look at AWS VPC for this.
Whilst VPC is free outside of the usual instance pricing, it doesn't work with Micro instances (the cheapest ones).
When not using VPC, you're assigned IP addresses through DHCP. When the DHCP lease expires, or you restart, your IP is released back to the pool.
VPC lets you use private IP addressing, you can use it with Elastic IPs and is much easier to integrate with a physical infrastructure setup.
If you're only testing/investigating AWS and have little or no budget to use anything other than a Micro instance, I'd just suck it up and deal with the changing of IPs.
If you've got a budget that lets you use instances other than Micro, then go for VPC.
Also, if you're doing more than testing/investigating I'd recommend starting with VPC straight away as trying to migrate from a non VPC to a VPC infrastructure is a massive PITA.