Hmm... Normally when pieces are appearing in strange places on the board, or you notice that a move looks different from when it did originally, there is an issue with a cached page. From what I can see in the game that is what took place. Sorry for the confusion!
Move Glitch

That makes sense; thank you for the answer! Is there any way to correct a cached page or prevent the issue with the cached page from happening, even if it is my opponent's move?

Sure, I can refresh the page when it is my move (thank you!), but the trouble is that when I saw my opponent in the game above make Bb6, it stayed that way on his computer after he hit "submit move." I logged on 30 minutes later to find Bb4+. There was no lag on his computer, and the Bishop moved to a different (yet legal) place on my computer. However, could it be possible that it was the computer that I was on that needed refreshing before I make my move, and would it place the Bishop back to Bb6?

We do suggest you clear your browsing cache occasionally. There are some instructions here http://support.chess.com/Troubleshooter/Step/View/2
If you let me know what browser and operating system you're using I can give more specific instructions :)

On the right side of your address bar, click the chrome menu button


Thank you for the instructions! So, I just empty the cookies and cache before I make a move to ensure that my opponent's move was the one he/she made?
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I was playing a correspondence chess game against a friend of mine, and he came to me with his laptop, and he let me see a move that he made. I saw him do "Bb6." I told him, "I got your Queen," because he never realized that I discovered-attacked his Queen by moving my Pawn to attack his Bishop and support my Knight. When I logged in to make my move, it was different than what I saw, and it put me in check. I had no choice but to block it. Since I told him that his Queen was in danger, he then moved it out of the way. I was very confused as to why his Bishop was at a different spot than where it really was in. The next day, he told me that he noticed the glitch as well. Here is what happened:
It did not matter, but I am mainly worried that it will happen again, for or against me. Is this a rare event, or does it happen from time to time, and can it be prevented?