Possible solutions to this question can be found in several places, including this forum post on the Minecraft Forum website. Here are a few things you should try if you are having this problem:
- Check to make sure you have the latest version of Java installed. To verify your Java version, you will need to use a browser other than Chrome. (As noted by Robotnik in the comments, skip this step if you have the latest version of Minecraft already installed)
- Change Minecraft's performance settings (check here for details on video settings) in Options > Video Settings:
- Set Mipmap Levels to OFF. Slide the menu bar or click on the far left side of the menu bar to set this to OFF.
- Set Render Distance to a lower chunk size, as this will load less chunks around your location.
- Change Graphics and Clouds to Fast (instead of Fancy) to lower the graphics.
- Set Smooth Lighting to Minimum.
- Change Particles to either Decreased (only half of all particle types appear) or Minimal (only the most important particles will appear).
- Change Use VSync to OFF and instead lower your Max Framerate. See the link discussing performance settings for more information about this and other settings.
- If you haven't already, update Minecraft. According to the forum post in the first link, the reappearance of broken blocks is a known bug for Minecraft versions before 1.8. The forum also suggests that the bug may be centralized around your spawn location, so if you do not wish to update, move about 50 blocks away from spawn.
Overall, checking Java and changing your Minecraft video settings should do the trick. Block reappearance usually means you're lagging and/or your graphics settings are too high for your computer.
Side Note: If you ever have this problem in Multiplayer, the problem may also lie with your internet connection or server plugins.