The least harsh penalty for plagiarism is that the professor gives you a failing grade for the assignment. If you are lucky, he may allow you to redo the assignment and give you a grade for it. Since plagiarism is covered in most syllabi and by the college itself, claiming ignorance will not work.
A professor can take the punishment a step further and bring the issue up to the judicial committee at the college. If you are found guilty of plagiarizing, the college judicial board will impose an appropriate penalty. Each school varies its rules. You may have your grade decreased by a full letter or automatically be given an F for the course. The fact that you failed due to plagiarism may also be reflected on your transcript.
You may also get suspended from school for plagiarizing content on a paper or other assignment. Suspension terms can vary from a week to a few weeks, the rest of the quarter or the rest of the semester. This may impact the grades in your current classes and your eligibility for financial aid for future semesters.
Expulsion is a final step for plagiarizing content. Most colleges will not expel a student automatically for plagiarizing but may do it after repeated incidents. Being expelled means that you are prohibited from attending the school at all. If you are a junior or senior, your punishment may be more strict because you have been at the school longer and should be aware of the various policies and consequences.