In American English, a student is anyone who studies. In terms of school, you are a student as long as you are taking classes! It is also a general term meaning someone who is learning or studying something. So, you can be a student of almost anything!
Some countries/languages use the word "pupil", who is someone that has not yet graduated high school. So, while you are in elementary (Kindergarten to 5th grade), middle (6th-8th grade), or high school (9th-12th grade), you are a pupil.
An undergraduate is a college student working towards a Bachelor's degree (4 years) or below (like an Associate's (2-year) or a certificate). Once you get your Bachelor's and you are still taking classes, you are a graduate student. Graduate students usually work towards Post-Baccalaureates, Master's Degrees, Doctorate Degrees, or even Post-Doctorate Degrees!
Undergraduate level classes are usually labeded 1xx-4xx, with anything over 499 being a graduate level class.
Then there is the case that a graduate student is taking undergraduate classes, meaning that they are taking classes for undergraduates, but they already have a degree. Similarly, undergraduates can take graduate-level classes, but they will still be undergraduates until they get a degree!