The Ethereum community has elected to do a hard fork in order to resolve the DAO exploit. The MainNet block on which the hard fork will occur is set at 1920000, which should be reached on July 20 or 21 depending on your time zone.
Please be very cautious with your private Ethereum wallets and do not send any transactions during the fork to anywhere else. Even after the fork you should be careful and understand the danger of the replay attack. If you only care about your funds on the dominant chain, then you do not need to worry about the replay attack.
But if you care about funds on both chains then you will need to take precautions against the replay attack. Unfortunately, aside from the highly technical guide linked above there isn’t a comprehensive guide for more ordinary users on how to protect yourself. But here is one place with some tips and discussion that is aimed at a somewhat less technical level:
https://www.reddit.com/r/ethereum/comments/4t9mzg/quick_hf_safety_tips_for_users/
Hopefully a better guide to navigating the fork with your private wallet will be available soon on https://blog.ethereum.org/ or elsewhere.