Custom Properties are essentially arbitrary tags with names meaningful to you. When you apply them to some resource, the tag becomes a property of the item (from Omea's point of view). You can see this most easily by the fact that you can sort the list of resources according to Custom Properties.
Extra Power for Views
Let's say you create a property called Reply Due By <date> and apply it to emails that need a reply by the date specified. Later, you can quickly pull all these messages together by searching for emails having the Reply Due By property, or by creating a new View containing only items that have the Reply Due By property.
Extra Power for Searches
When you use Advanced Search, you can include the custom property or several properties as condition or exception, providing an additional layer of granularity in the search.