There are many different types of military hats that were worn throughout history. Some had foreign names and others had different names for different types. Here are a few.
Helmet
visor cap
overseas cap
busby
shako
pickelhaulbe
steelhaulbe
pith helmet or sun helmet
crash helmet or flight helmet
a pilgrim hat is called a pilly hat
K-Bar
The collapsible hat, very common in 19th century theatres, was called an OREMA.
beret
A hatter
Ushanka
A military hat is called a cover because it literally covers and protects your head. It's simply just a more formal term for a hat. (i.e. guns vs weapons, bathrooms vs latrines, etc.)
A military hat with a turned-up brim is commonly referred to as a "campaign hat." This type of hat is characterized by its wide, stiff brim and is often associated with various military and outdoor organizations. The design provides protection from the elements while maintaining a distinctive appearance. Campaign hats are often seen in contexts like scouting and military uniforms.
It's called a kepi - pronounced kay-pee.
A cadet's hat is commonly referred to as a "barracks cap" or "garrison cap." It is a type of military headgear typically worn by cadets and service members, characterized by its folded design and lack of a visor. In some military academies, it may also be called a "cover." The style and insignia can vary based on the specific branch of the military or organization.
Mussolini's Territorial Army, called "Milizia" (Militia) wore a black fez.
cap and gown. the hat is called a mortarboard.
a leperchaun's hat :P
It's called a homburg hat
Its called a hat box
According to Irish folklore, it's called a cocked hat.
A guardsman hat is called 'Bear Skin'