What is the purpose of the lost+found folder in Linux and Unix?
If you run fsck, the filesystem check and repair command, it might find data fragments that are not referenced anywhere in the filesystem. In particular, fsck might find data that looks like a complete file but doesn't have a name on the system — an inode with no corresponding file name. This data is still using up space, but it isn't accessible by any normal means.
If you tell fsck to repair the filesystem, it will turn these almost-deleted files back into files. The thing is, the file had a name and location once, but that information is no longer available. So fsck deposits the file in a specific directory, called lost+found (after lost and found property).
Files that appear in lost+found are typically files that were already unlinked (i.e. their name had been erased) but still opened by some process (so the data wasn't erased yet) when the system halted suddenly (kernel panic or power failure). If that's all that happened, these files were slated for deletion anyway, you don't need to care about them.
Files can also appear in lost+found because the filesystem was in an inconsistent state due to a software or hardware bug. If that's the case, it's a way for you to find files that were lost but that the system repair managed to salvage. The files may or may not contain useful data, and even if they do they may be incomplete or out of date; it all depends how bad the filesystem damage was.
On many filesystems, the lost+found directory is a bit special because it preallocates a bit of space for fsck to deposit files there. (The space isn't for the file data, which fsck leaves in place; it's for the directory entries which fsck has to make up.) If you accidentally delete lost+found, don't re-create it with mkdir, use mklost+found if available.
The lost+found directory (not Lost+Found) is a construct used by fsck when there is damage to the filesystem (not to the hardware device, but to the fs). Files that would normally be lost because of directory corruption would be linked in that filesystem's lost+found directory by inode number. Some of these might be lost directories or lost files or even lost devices. Each filesystem should have its own lost+found directory, but you might be looking at a system with only one filesystem. In general, you should hope that the directory is empty; but if there is corruption, be thankful that in many conditions files can be recovered after fsck places them here.
From "Linux Filesystem Hierarchy", section /lost+found":
As was explained earlier during the overview of the FSSTND, Linux should always go through a proper shutdown. Sometimes your system might crash or a power failure might take the machine down. Either way, at the next boot, a lengthy filesystem check using fsck will be done. Fsck will go through the system and try to recover any corrupt files that it finds. The result of this recovery operation will be placed in this directory. The files recovered are not likely to be complete or make much sense but there always is a chance that something worthwhile is recovered. Each partition has its own lost+found directory. If you find files in there, try to move them back to their original location. If you find something like a broken symbolic link to 'file', you have to reinstall the file/s from the corresponding RPM, since your file system got damaged so badly that the files were mutilated beyond recognition. Below is an example of a /lost+found directory. As you can see, the vast majority of files contained here are in actual fact sockets. As for the rest of the other files they were found to be damaged system files and personal files. These files were not able to be recovered.