In the NCBI Taxonomy tree what does "no rank" mean?
Well, welcome to Phylogenetic Systematics.

The Linnaean Taxonomy is a rank-based classification system. When Linnaeus first proposed it in 1735, in his famous Systema Naturae, he created 5 ranks: Regnum, Classis, Ordo, Genus and Species (in Latin). 

Today (more on that below) Linnean Taxonomy uses 8 main ranks:


Domain
Kingdom
Phylum
Class
Order
Family
Genus
Species


If you add the prefixes super-, sub- and infra- to that, you get more ranks.

However, in the 60's, a new systematics was proposed by Hennig, called Phylogenetic Systematics. I'll not explain in details what Phylogenetic Systematics is in this answer, there are plenty of good sources for you to read about. My point here is only one: showing you that there is no rank in Phylogenetic Systematics.

Have a look at this cladogram:



The group made by A + B is a monophyletic group. The group made by A + B + C + D is also a monophyletic group. The same way, the group made by A + B + C + D + E + F + G + H is a monophyletic group (by the way, there are 7 monophyletic groups in that cladogram). And all those groups have names.

Now comes the important part: in Phylogenetic Systematics, the number of monophyletic groups between the more exclusive set (you) and the more inclusive set (your very first ancestor, more than 3 billion years ago) is enormous! It is, of course, way bigger than 8 (or than 32, if you add all those prefixes). Even if you add uncommon ranks, as Tribe  or Section, you don't get even close to the number of monophyletic groups between you and your oldest ancestor.

As I read some time ago in a phylogenetic systematics paper: "We need names. Lots of names!". 

That's why modern taxonomy has a problem with the traditional notion of ranks, some biologists defending that we should drop the ranks altogether (except, of course, the Species). And that's why I was uncomfortable in writing today some paragraphs above: there is an increasingly number of biologists that use phylogenetic nomenclature, which is incompatible with Linnaean taxonomy.

So, to give you an example: Mammalia is a Class. However, Mammalia lies inside a higher group, that we call Synapsida. Synapsida is neither a Class nor an Order nor a Phylum... it does not have a rank. Synapsida, by its turn, lies inside Amniota (animals with amnion), which also doesn't have a rank...

The examples abound: Sauropsida, Eudycots, Opisthokonta, Reptiliomorpha, Viridiplantae etc are all groups without a rank.

Thus, when a biologist says "that organism is an Amniota", she/he is just saying that the given organism belongs to the monophyletic group called Amniota. If you ask him/her "what is Amniota? A Class, an Order, a Superfamily, a Subphylum?", the answer will be: "none of them". Monophyletic groups don't need to have a rank in Phylogenetic Systematics.

Finally, quoting the Taxonomy Help from NCBI:


  The ‘preferred scientific name’ for a taxon may be either formal (e.g. Homo sapiens) or informal (e.g. Homo sp. Altai, or uncultured fungus). Taxa above the species level may also be either formal (e.g. the order Mammalia) or informal (e.g. the unranked node eudicotyledons). (Emphasis mine)


Source: https://www.ncbi.nlm.nih.gov/books/NBK53758/#taxonomyqs.Overview