In the United States, an undergraduate or an "undergrad" is a student pursuing a bachelor's degree program (usually 4 years) at a college or university, while a graduate or a "grad student" is a student who, having obtained a bachelor's degree, is now pursuing a master's or "graduate degree" in graduate school.
These concepts are often confused, because the terminology in the rest of the world is different. What's known as undergraduate studies in the US is referred to as graduate studies in most other countries. And the equivalent for graduates (graduate students, graduate studies) in the US are known as post-graduates (or post-graduate students, post-graduate studies) in the rest of the world.
edit Video Explaining the Differences