You will not get charged for sending a text message from the web site or by sending an e-mail to email@example.com. However, the person receiving the text message will more than likely be charged (or have it deducted from their messaging plan).
The only problem with texting from AT&T's website is that the replies get sent to your phone. Those replies are deducted from your messaging plan or charged as pay per use if you don't have a messaging plan.
You can text easily and FREELY from the computer if you want to text while you're on your PC.