Applying for a credit card is something you should think about beforehand, rather than afterward, but it's okay if you have second thoughts after you're approved. Even if the card is issued, there's no obligation for you to use it, especially if you see something you don't like in the user agreement. Cancelling one is a pretty straightforward process, though you should always follow up to make sure the account was actually closed.
Start With the Phone Call
Your card usually arrives with instructions to sign it and activate it before it's used, but that doesn't mean the account isn't "live." If you just cut up the card and toss it in the trash, the account is likely to show on your credit history for years. Instead, pick up the phone and call the card issuer's toll-free number for customer service. You can expect the card issuer's representative to put some effort into trying to keep you as a customer — they approved you for a reason, after all — but if you stick to your guns, you'll get what you're asking for. If the card charges fees to open an account, you might be expected to settle those; they're intended in part to defray the cost of checking your credit and issuing the card, but you should certainly try to get them waived. As always, when you're making this kind of call, keep detailed notes of which agent you spoke to and what terms were agreed upon.
Follow Up by Mail
If you're serious about doing this properly, you should follow up by mail. That's especially the case if you negotiated a specific closing date, or a waiver of fees, with the agent you spoke to. Cite the agent's name and — if applicable — number, the date you spoke and everything that was agreed to in the course of your phone call. Send it off with a return receipt requested, so you can prove that your letter was received by the card issuer if it should become a problem. Keep a copy of the letter and the mailing receipts, as well as any other pertinent documentation, until the account shows as "closed" on your credit report. You may still get statements for the card for a few months afterward, but that's perfectly normal — don't panic unless the account shows a balance other than zero.
Question Your Motives
Before you cancel that card, though, it's worth taking a moment to ask yourself whether it's in your best interest. If you applied on a whim, and don't really need the card, that's a valid but not entirely compelling reason to cancel it. Unless your shiny new card packs some serious fees or a ridiculous interest rate, or you have a history of getting into debt trouble, there's no particular reason to deep-six it. Having the additional card won't hurt your credit — the application itself causes a dip in your score, but a very small one — and can actually be a positive.
When It's a Good Thing
The big reason you might want to keep that card, even if you never use it, is a thing called your "credit utilization rate." The short version is that lenders look at how much potential credit you hold, and how much you're actually using. Suppose you have only one card, with a $2,000 limit, and you have a balance of $1,000. Your utilization rate is 50 percent, which is quite high. Now, suppose you have five of those cards, totaling $10,000 in available credit, and the same $1,000 balance. This drops your utilization rate to a much more reasonable 10 percent, making you look better to lenders. If there's nothing wrong with the card itself and you're dropping it just because you don't need it, you might be better off keeping it so your utilization rate stays lower.