Take for example the human and the chimpanzee, they are "closely related" species since they are "close" to each other in a phylogenetic tree. However, this terminology seems pretty informal, what would be the formal way to specify that two or more species are "closely related"?
Are you looking for something like "sister lineage" (or "sister taxon" or "sister group")
This term may be more specific than what you're looking for, and to use it correctly you'd want to be a little careful – I guess in this case you might have to say something like:
"the lineage that includes chimpanzees and bonobos is sister to the lineage of which humans are the only extant species"
"the lineage that includes chimpanzees and bonobos and the lineage of which humans are the only extant species are sister lineages"
both referring to this phylogeny.
It's also possible that explaining exactly what you mean by 'close', or what it entails, would be appropriate, be it Ilan's suggestion of % genetic similarity (or maybe genomic similarity?), # of nodes between your two taxa (again, this would have to be in reference to a specific phylogeny), or if you have a dated phylogeny you could refer to the time since divergence, as in "coding DNA divergencies ...place the most recent common ancestor of humans and chimpanzees at between 5 and 6 million years ago."(Wildman et al 2003).
As genus comes above species I'd consider using this term for the species which are close genetically ("these two species are from the same genus"). Above genus comes family etc...
On the other hand, the proximity or similarity of species could be probably "measured" by degree of biodiversity, and especially genetic diversity considering two close species having low biodiversity and very different species as having higher biodiversity.
Thirdly, and I feel that this is the most strong term to use, you can "express" the genetic closeness of species as being "genetically similar":
"Some monkeys and humans have 98% genetic similarity" or "are up to 98% genetically similar" or "... DNA similar"...