A SMART hard drive is a hard drive that includes built in software that checks the drive for problems. The software runs at the firmware level and is running underneath Windows. It will detect problems when Windows will not run.
SMART stands for Self-Monitoring, Analysis and Reporting Technology. The drive is monitoring itself for problems with the drive and reports them to the person using the computer. SMART is reporting problems actually in the drive not that files are messed up. SMART will tell you when areas are going bad on the dive.
Usually if the drive begins reporting problems it is telling you that the drive is about to fail. If you get problems reported by SMART you need to immediately backup anything on that drive you want to keep (hopefully you regularly backup the information anyway) to another drive (whether USB thumb drive or USB external drive. Then you need to get the dive replaced, or if it is an older machine decide if you may be better off buying a new machine.