I agree with everyone else that it is ideal to pay for application expenses in cash as much as possible. But for some of us, a credit card is a necessary reality to pay for this ridiculously expensive process. If you are smart about managing your debt, it is not the end of the world. I work full-time already, but there is no way I could have come up with the thousands of dollars I have spent so far on applications and interview expenses on my own, which is what I would have to do--my mom is currently unemployed and even when she had a job, she made less than I do (no father involved). If you have zero family support and have to find a way to do this on your own along with paying for all your other expenses (rent, food, medical care, etc), a credit card is one possible (if less than ideal) solution.
Some tips for the OP:
-As others have mentioned, use the card only for expenses you can't cover yourself. In other words, don't get a credit card to pay for med school applications so you don't have to get a job or so you can take more vacations. Exhaust your other options (your earnings, savings, and family resources if you have them) before turning to a card. Consider delaying your application to another year to save money (I didn't want to do this, which is why I decided to get my first credit card this year so I wouldn't have to wait to apply. But if you're not in a hurry, it could be a better way to go).
-If you do decide to get a card, make sure to read the fine print--some of these cards limit the 0% APR to certain things or have other strings attached, but there are some good deals out there. I got my first credit card specifically to help pay for this process, one with a 6 month 0% introductory APR on all purchases and balances. When the 6 months expired, I switched my balance to another card, one with a 0% APR that doesn't expire until 2006. I hope to have it paid off by then, but if not it is always possible to switch again. In this way, you avoid racking up interest charges, which is how most people end up with credit card debt they can't manage. Pay attention to the deadlines on special offers like 0% APR so you don't end up with high interest charges.
-Use the card ONLY for medical school expenses you can't cover yourself if that's the reason you're getting it. It is easy to get into the mentality that you're already spending so much or have so much debt, why not charge a few hundred for something you'll really enjoy? This is dangerous and will lead to trouble. Decide what you're going to use the card for, and only use it for those things. This will limit your debt.
-Pay more than the minimum balance each month. This is critical if you are getting interest charges, but even if you have a 0% APR on your card if you only pay the minimum balance you will basically never get the card paid off. I generally pay 4-5 times the minimum each month.
Being poor sucks, but if this is the only way you can pursue your dream don't feel guilty about it--just be smart so you don't end up with debt you can't manage. Good luck!
Edit: I just re-read your post and saw that you already have a credit card now for "personal use." Either pay that one off and get rid of it before getting another one, or use the one you have now if you can manage paying off the charges you will put on it, but I would advise against having more than one card at a time as it is too easy for things to get out of hand that way.
Also, just to clarify, I am NOT suggesting that the OP get a card and run up debt he/she has no way to pay off. Rather, I am suggesting that a 0% APR card, if used carefully, can be a better solution for those who lack other sources of funding than, for example, a personal loan, which will immediately begin accruing interest and will have to be paid back right away on the bank's terms. A card with a good introductory offer, such as a 0% first year APR, can essentially be viewed as a 0% one year loan that offers flexibility in the monthly payment schedule; this is a good thing as long as you have a plan to pay it off before starting school when you will be too poor to manage additional payments. If you are a student, it would probably be better to talk to your fin aid office about getting more funding from them, as most student loans are low interest and won't have to be paid off until you are finished with school and in a better position to do so.