Every time you start up an instance in the cloud, you will be provided with a hostname and IP address. If you shutdown your instance, you could get a new hostname and IP address. Most of the time for cloud servers, the name and IP address doesn't matter and the possibility of it changing won't affect your work. However, if you need to guarantee that you have the same IP address and hostname every time your instance is restarted, you need an elastic IP.
To add an elastic IP address on your instance, in the Eucalyptus console, click on Instances. Then click on the ... under the Actions column next to your instance name. Choose "Associate elastic IP address with instance" and select whatever IP address is offered in the drop down list. Then click Associate Address with Instance. That is now a permanent IP address you can use to refer to your instance.