It has a single endpoint
I don't really care what else is going on in your API, any "RESTful" API which has a statement such as "all requests are made to http://example.com/rest" is ... not RESTful. REST is all about handling representations of resources, each is represented by its own URI and we operate directly on that. If it looks like "pretty URLs", then it's probably along the right lines.
All requests are POSTs
While it is completely valid to have POST requests in a RESTful service, if all requests to this API are to be made using POST, it's very unlikely to be RESTful. POST requests create new resources, so if that's all that is happening, then great. If you're able to retrieve, update or delete data, then this isn't a RESTful service.
Response metadata is in the body, not header
Any sign of data along the lines of
status = success in the body of the response again means that this isn't RESTful. REST takes advantage of the HTTP envelope format and puts all the supporting metadata into the headers of the requests and responses. The body of the representation is literally just the representation of the resource.
There are verbs in the URL
This can be quite subtle but shows up especially with subresources. Look out for something like
/item/42/activate ... REST deals with resources, an "activate" isn't a resource. If you're modelling a service, maybe something like
/item/42/status could make a better alternative.
The URL includes method names
This is the easiest "tell" by miles. The most obvious non-RESTful aspect is when the URL has something like
?action=getRecentItems in the URL. Sometimes, this clue gets combined with another one, and the method name will even be in the body of the message - definitely not very RESTful!
Do We Care About RESTfulness?
Personally, I think that how useful an API is, and how RESTful it is, are entirely uncorrelated. Do take care though when describing your "sends-data-over-HTTP" application as "RESTful", because chances are, it might not be. Look out for anything with the word "REST" in it (if you ask nicely I might share the comments that I don't publish on this post) because REST has very specific criteria that can sometimes be difficult to adopt, and for some reason people interested in this subject seem to like to tell other people they are wrong a lot!