I believe the instance goes away. If you've persisted data to EBS, you can attach the volume to a different instance. Anything outside of EBS is lost. However, if you created an AMI (AWS Machine Image), you can launch a new instance from that. (but it's a snapshot, not a live image, so it'll be only as up to date as it was when saved)
Some virtualization platforms support live migrations (hyper-v and ESXi), but this isn’t something AWS supports (afaik). Anytime your instance is stopped/started or rebooted then you may end up on a different physical host.