Wave was marketed as a successor to e-mail, but (1) that just confused everyone (because what does "successor to e-mail" even mean) and (2) in my experience that was not a good way to view it.
A simpler way to view Wave was as a federated form of Google Docs with added support for threaded conversations in the style of e-mail threads or chat logs.
If we want to break it down, Wave is/was a federated, personal, realtime, wysiwyg wiki with strong support for threaded conversations, sharing, history, and privacy and access controls. This combination of features means that wave can be used for a wide variety of communications from chat to e-mail to blogging to collaborative document writing.
Wiki: I'm going to assume you are already familiar with the "Wiki Way". Basically a "wave" document is a wiki page that you create and can freely edit.
Threaded: While wave allowed multiple users to edit document as holistic entities, it also supported structuring documents as a mailing-list style tree structure of threaded replies. Private side conversations not visible to the entire group were also supported.
Realtime: Perhaps the most important feature of Wave was that your edits were instantly viewable. If you and a friend make successive edits, it can be used as a chat system. (Wave had support to make this easy and so each additional edit could be appended at the end.)
Personal: The wiki documents that you create exist in their own private namespace (e.g. just like documents on different webservers) and usually (always?) are just internal identifiers not intended for to directly manipulate. This avoids the problem that standard wikis have of dealing with contention for document names.
Sharing: You could send a "wave" document to another user and have it appear in their inbox. You could also invite others to participate in an existing wave. Your inbox would also be notified when a wave that you were watching was updated (e.g. like how you receive a reply to an e-mail).
Privacy: Wave documents are by default not publicly visible and have sophisticated and easy to use access controls.
WYSIWYG: User's edit their documents as they would in any modern word processor or e-mail client without having to know about any sort of wiki syntax.
History: Like any good wiki, wave supported viewing older versions of a document.
Federated: I can run a wave server and you can run a wave server and both of them can interoperate. This is similar to how different companies can run different e-mail servers and unlike private communication systems like Facebook messaging. For people who want to maintain a free internet, this is an important feature. However, unlike e-mail, the authoritative copy of the document stays on the creators host. Though there is support for caching, I never clearly understood where wave fell on the problem of hosts lying and trying to rewrite history.
In the end, Wave was a good product that was marketed poorly (the marketing explanations left people not knowing what it was). The efforts to make it federated required the creation of a public spec which also probably slowed down development.
I think Wave was an awesome idea and others should build on its concepts, but shouldn't tie themselves to how those ideas were implemented in wave. In particular, the rise of Facebook may have changed what people expect from their communication platforms so some of Wave's ideas may need to be updated to encompas that. (I don't use Facebook so I don't know what those changes would be.) But if the next social platform were to support Wave-like ideas, then it could be very nicely positioned as a Facebook-killer as it could represent a next-generation advance in social platforms.