In the United States, the terms usually refer to university education. I'll illustrate the most common use of the terms by describing my own academic career.
I graduated from high school when I was eighteen years old. I went to a public high school. My parents did not pay tuition. The school was supported by tax money and was free to everybody.
I entered college, which in the US means the first four years of university education. It was a private university. My parents paid a lot of tuition. I studied for four years. During that time I was an undergraduate. I majored in mathematics. I graduated with a bachelor's degree. My diploma says that I hold a "Bachelor of Science in Mathematics."
I then went to graduate school, at a different university. It was a state university, a public university. I had to pay tuition, but it was very low. I was a graduate student, or "grad student." I was doing postgraduate work.
In the verb "to graduate," as in "I graduated," the "a" is pronounced as a "long A." I GRAD-you-ATE-ed.
In the noun, "I'm a college graduate," or adjective, "a graduate student," it is pronounced as a schwa, something like "GRAD-you-it," but you hardly pronounce the vowel sound at all.