I take it you are referring to the drive menu on windows installer, delete basically puts the drive partition to unformatted raw mode which is unusable until formatted again (Mainly used if you had multiple partitions that needed put them back to one drive again for whatever reason) Format sets the partition ready to be used for windows, clears all previous data from the drive and does the basic partition settings like the boot information and file system (NTFS) etc. Quick format is as it states is a quick format which merely clears the file headers and marks any data to be wiped without checking the entire partition, hence much quicker but won't see any errors.
Format and quick format both write the directory entries for a drive recording all the sectors available for use on the drive. Format additionally runs a check of each and every one of those sectors confirming that each one is readable, quick format does not so problems may exist on the drive which will not be detected by quick format. Delete if you're referring to file delete, does not write the directory entries at all, it simply sets the entries for the file(s) being deleted to empty so that all of the sectors which they occupied show as being available. Deleting a logical drive of partition releases the sectors allocated to the drive and shows them as unallocated. None of these processes, format, quick format or delete (partitions or files) physically removes any data from the drive and it is quite easily recoverable to anyone with the necessary skills and tools.
A full format on a 1.5 Terra Byte drive will take a few hours 3 plus in some cases. Many users recommend doing a full format as it checks for bad sectors and isolates them from being used. Even a new HD can have a bad sector or two. No you need a program that writes x and 0's to the sector for more than one pass usually 3 is recommended . Here is one I noticed in PC World http://www.pcworld.com/downloads/file/fid,157500/description.html?tk=nl_ddx_h_dlfeat actually CCleaner can do this also under driver wiper DOD standard is 3 passes NSA standard is 7 passes. On a large drive this will take a few hours.