Quick format on an NTFS drive means re-initializing the MFT's (the equivalent of FATs)
A full format keeps the disks file system in consideration and formats the drive exactly according to the specifications of the file system. After a full format, the drive/partition is ready for use.
A low level format on the other hand doesnt take into consideration the file system and totally erases all traces of data (including the FAT/MFT, MBRs, etc). This makes the drive appear as if it were a freshly manufactured disk.