How you define cheating will play the biggest part in how you ultimately answer that question. For example, some people would only consider it cheating if physical contact is made, while others would say emotional cheating also counts. There may be some gray areas, but your feelings should let you know whether you have been cheated on or not.
If you catch your partner spending hours and hours chatting with the same person online, then it's only logical that you would be suspicious. How much you trust your partner will play a key role in how you respond. A partner that quickly turns off the computer screen or tries to hide their messages would be enough to raise the alarm for anybody. On the other hand, if they continue chatting online even after you come into the room, and they don't mind you reading their conversation, then it's a sign that you don't have a lot to worry about.
Of course there can be more to an online relationship than seemingly harmless text chats. You also need to look at the contact of those chats. Any sexual talk could be nothing more than simple flirting, or it could be more. However, if your partner is sharing deep, dark secrets, then it shows that the online relationship is more than superficial.
Confronting your partner about this won't be easy, but it has to be done. Your initial approach could be one of simply trying to understand exactly what is going on. However, you need to be aware that they may minimize what they are doing. Right now, your goal is to get to the truth.
Don't be surprised if your partner tries to make excuses, denies that they are cheating, or comes up with justifications for their behavior. They may be trying to convince you that what they are doing is okay, or they may be trying to convince themselves. Either way, it's up to you to make it clear that what they are doing is not acceptable.
It may be hard for your partner to understand how they could be cheating if no physical contact was made, but be patient and explain how they were still cheating. Once they understand it, they should be willing to admit that they were cheating, and then respond appropriately.
Are online relationships cheating? They sure can be, and infidelity in the virtual can hurt just as much as infidelity in the real world. How you proceed from here will depend on many factors, but always remember that it is possible to heal and move on.