In morphology (a subfield of linguistics), a compound is a (relatively) new word created by the word formation process of (surprise!) compounding. Usually, compound is short for compound noun, because the compound word is a noun, though the other word may be of any lexical category.

How do compounds differ from noun phrases?
Compounds are any combination of words (generally two in English) that have come to be represented as one word in the mental lexicon of the speakers, unlike noun phrases, which are thought of as a combination of multiple words which can be switched around (i.e., noun phrases are productive.) The same words can be used as a compound noun or a noun phrase (see examples below), so telling the difference between these two things is often difficult (and something a number of my Introduction to Linguistics students always get wrong) when the compound is a combination of an adjective and a noun. When you have a noun-noun compound (like enchilada sauce) they are easier to identify, but still must pass the intonation test below.

This is all well and good, but how can we identify a compound?
The testable difference between compounds and noun phrases in English, is which part of the word or phrase is stressed. Rather than the head receiving stress, as happens in phrases (we say black bird because it is a bird, not a black), the first item receives stress, regardless of whether it is the head or not (and it usually is not, because of how English syntax works).

For example:
black bird = could be a raven, crow, blackbird (noun phrase)
vs.
blackbird = a specific species of bird (compound)

white house = any house that is white in color
vs.
White House = where the president of the united states lives

make up = a verb phrase meaning to create something (as a story), to prepare something for use (as a bed), or to make amends after a fight
vs.
makeup = cosmetic products or
make-up = the composition of something

(Note that the spelling is generally different between a noun phrase and its compound counterpart; this in and of itself, however, does not tell us whether a word is a noun phrase or compound noun, as in makeup and make-up which are both compounds but have different meanings. Spelling is often altered in situations like these because readers do not have the benefit of intonation to help their understanding of a sentence, but linguists are much more interested in how the language is spoken, not how it is written. Spoken language reveals much more about how language works than do things like spelling conventions.)