pen vs ballpoint

pen and ballpoint both are nouns.

Word	Noun	Adjective	Verb	Adverb
pen	Yes	No	Yes	No
ballpoint	Yes	No	No	No

As nouns, ballpoint is a hyponym of pen; that is, ballpoint is a word with a more specific, narrower meaning than pen:

pen: a writing implement with a point from which ink flows
ballpoint: a pen that has a small metal ball as the point of transfer of ink to paper

pen (noun)	ballpoint (noun)
female swan	a pen that has a small metal ball as the point of transfer of ink to paper
a writing implement with a point from which ink flows
an enclosure for confining livestock
a correctional institution for those convicted of major crimes
a portable enclosure in which babies may be left to play

pen (verb)	ballpoint (verb)
produce a literary work