If you've had a "student" credit card for at least two years, or if you have other forms of credit - such as a student loan or perhaps an auto loan - you may have built up a sufficient payment track record that would qualify you for a traditional credit card geared toward any adult.
Student credit card offers are primarily designed for people in college who have limited or no credit history. In the wake of credit card reform legislation, i.e. the Credit CARD Act of 2009, issuers of student credit cards now also require you to have a co-signer if you're under 21 years of age. Otherwise, to obtain the credit card on your own you have to prove that you have an income of your own that would allow you to repay what you could charge on a given credit card.
The only sure-fire way to know if you'd qualify for one of these "regular" cards is to apply. Of course, there's another fairly obvious turning point that will tell you to that's it time to graduate from your student card: it's once you yourself have left school or graduated.
At that point, you're entering the "real world" and your status as a student is less important or relevant to credit card issuers. So they'll start evaluating your creditworthiness just as they judge anyone else applying for a credit card.
Regardless of your status, the basic fundamentals of credit use apply: don't borrow more than you can afford to pay back, and make all your payments on time, every time.