There are numerous Carpinus species around the world that are listed as hornbeam in one form or another, but in the USA the name primarily refers to C. caroliniana, of the family Betulaceae. There are numerous other common names that are used and I'll add more about that later, but for now, I point out that they include blue beech and ironwood as the most commonly used ones in the USA.
Like its close relative hophornbeam, this wood is very hard and tough (hophornbeam is a bit more) and it is also (like hophornbeam) known as ironwood in the USA. Ironwood is one of the most useless common names in existance in terms of actually identifying a wood --- I have what I'm sure is an incomplete list and it shows 180+ different species that have ironwood as one of their common names.
my samples:
small piece contributed by Iain Rankin, whom I thank for this and other contributions to the site
also provided by Iain Rankin, this is a pic of a spindle gouge handle he turned for an antique Sorby tool.