A counter offer can be a delicate situation and requires careful consideration.
Examine your initial reasons for wanting to make a change; often the reasons people make job changes are for issues other than money. If this is the case, then it is likely you will return to those same issues if you accept a counter offer, after the initial glow of more money and feeling appreciated by your current company wears off. On the other hand, if money, or not feeling appreciated as a result thereof, was the primary reason for making a change, you might be happy with accepting the counter offer. It is a good idea to list out the pros and cons for each opportunity and discuss these with someone whose opinion you value.
There are always some risks in going into a new position with a new organization; however, there are also risks in accepting a counter offer. Depending upon the relationship you have with your manager and/or management team, and the corporate culture (values, attitudes, etc.) at your present company, accepting a counter offer could change how you are viewed. There is the possibility of being seen as disloyal, and if the outside offer came at a very crucial time — say, when losing you would have been disastrous to a vital project or the bottom line — you may cause some animosity if the employer feels there is no choice but to counter offer to keep you on board. These feelings could pass in time, but it is also possible for you to be targeted for replacement (or passed over for promotion, important projects, etc.) at a time when it is more convenient for your current employer.
Remember, this answer assumes you have not yet accepted the offer from the new employer, and your current employer, learning of your consideration of the offer, makes you a counter offer. If you have already accepted an offer from the new employer, it is often considered somewhat unethical to withdraw your acceptance based upon a counter offer from your current employer; however, you still have to do what is right for you.
In the end, after weighing all the factors and perhaps discussing them with family members, close friends or a mentor, you will need to make a decision. Ultimately, you need to do what is in your best short- and long-term interests. And usually, what is appropriate for one party is appropriate for both parties concerned — even if not always apparent at first.