Is there a term for simply "a unit of language"?
The term Constituent marks a particular kind of syntactic unit, one that has a coherent structure. It doesn't apply to just any string of words.
Constituents include sentences, clauses, and various kinds of phrases, like noun phrase, verb phrase, and prepositional phrases. They can be nested inside one another -- a sentence can contain a clause that can contain a verb phrase that can contain a noun phrase, for instance -- and these relations are sometimes sketched with brackets or with tree diagrams. It can get complicated.
For instance, the sentence
- The old man in the raincoat lit a cigarette.
has the following major constituents (all italicized):
the subject noun phrase the old man in the raincoat,
containing the noun phrase the old man,
and the prepositional phrase in the raincoat,
containing the noun phrase the raincoat.
the verb phrase lit a cigarette,
containing the object noun phrase a cigarette.
Phrase is the term used for a constituent that is not a clause (i.e, no subject plus verb). Any other string of words that go together is just a string of words, unless it's a constituent of a sentence. For instance, the following are not constituents of that sentence:
- old man in, in the, lit a, old man lit, the old, ...
All individual words are constituents, but it's the combinations that make the difference. The most important thing to know about constituents is that all syntactic rules apply to constituents only, so you can't tell how the sentence is constructed without finding the constituents.
There are hundreds of syntactic rules that are used to make tests for constituents, but I'm not gonna get into that here. See McCawley 1998 on tests for constituency.