When you select "Verify" you are comparing the original data bit for bit to the disc copy. Verification is an extremely high standard that doubles the time to burn. If it is not selected discs will still reject if there is a problem that makes it unplayable.
What does it mean when a disc fails verification or rejects during a normal burn?
Rejection during normal burning = disc burning is not complete. For a data disc, it probably will not open. For video, it is only playable to a certain point.
Verification Failure = one or more bad bits of information was found on the disc. The disc is still playable. For video, it could mean there is a missing pixel on one frame of one second of the video. For data, it could mean that one file has some corruption that may or may not cause it to open. Or the file could open but with some display corruption. For an installation disc, if one file is corrupt it may mean the entire installation would fail.
Verification is typically not necessary for video, is optional for data discs and is essential for software installation discs.
Increased verification failures could be the first sign that your disc drive is starting to fail.