Basically it's a set of rules, on how Mac OS X works.

Then there is the master set of rules, which has a better integrity, so when you verify permissions, it compares both sets, and if there are any differences you are told.

Then you repair. It simply fixes the differences. Some differences are normal.

Go to Applications>Utilities>Disk Utility and select the hard disk. Then select "Verify permissions", and then "Repair permissions".

EDIT: The main benefit is a good system. If rules go crazy, then you get odd things like the wrong Finder window opening when you try to open a folder. This is a rare problem though. It's just routine maintenance; some people repair once a month, others once a day.
01. 'What are permissions in OS X?' - Well, first - welcome to the wacky world of UNIX; and, see here about permissions.

02. '... what would be the benefit of repairing them' - Well, MacOS X (10.0.0 - 10.4.2) itself, at times likes to change the permissions of files (for unknown reasons); plus, you may have created or received a (document or application) file or files, and / or folders - whose permission(s) settings are not correct.

Then there are those times when you either re-install your MacOS X altogether, and / or install a related 'Update', 'ComboUpdate', or 'SecurityUpdate', etc. package. Prior to performing such an installation, it is now a common ritual to launch 'Disk Utility' and select 'Disk Repair', followed by selecting 'Repair Disk Permissions'. Then, once the new installation is completed - you are to once again launch 'Disk Utility' and select 'Repair Disk Permissions'.
Failure to perform such steps can result in an application not launching, etc.

Unlike with System 9.2.2 and earlier - where 'routine maintenance' was practically non-existent; with MacOS X it is now a monthly, weekly, or even (for some) a daily necessity!

Should you want the very OS that changes your permissions - to eh, fix those improperly changed permission settings - as well as, the permission settings of other files and / or folders, then consider this process.

I know, I know; and, you thought you purchased a Mac.
UNIX is the underpin to all this:


at the prompt type:
ls -al

You get something like the following:

drwxr-xr-x 39 movies movies 1326 23 Aug 14:04 .
drwxrwxr-t 7 root admin 238 11 Feb 2005 ..
-rw-r--r-- 1 movies movies 3 23 Oct 2004 .CFUserTextEncoding
-rw-r--r-- 1 movies movies 15364 23 Aug 16:10 .DS_Store
drwx------ 2 movies movies 68 29 Nov 2004 .DV Backup HD Store
-rw-r--r-- 1 movies movies 2342 14 Jun 08:53 .HlwV_options
drwxr-xr-x 2 movies movies 68 23 Aug 14:04 .MacOSX
-rw-r--r-- 1 movies movies 1 3 Mar 16:25 .OpenTerminalHere
drwx------ 2 movies movies 68 23 Aug 16:11 .Trash
-rw------- 1 movies movies 0 20 Aug 09:39 .Xauthority
-rw------- 1 movies movies 8004 23 Aug 16:07 .bash_history
drwxr-xr-x 6 movies movies 204 28 Jan 2005 .datastudio
drwxr-xr-x 3 movies movies 102 18 Jan 2005 .dbvis
drwxr-xr-x 2 movies movies 68 1 Dec 2004 .java
drwxr-xr-x 4 movies movies 136 1 Dec 2004 .jpi_cache
-rw-r--r-- 1 movies movies 25 10 Feb 2005 .lpoptions
-rw-r--r-- 1 movies movies 6 18 Jan 2005 .mpdt23
drwxr-xr-x 3 movies movies 102 30 May 22:58 .mplayer
drwxr-xr-x 4 movies movies 136 13 Mar 21:28 .mysqlgui
-rw------- 1 movies movies 1024 19 Feb 2005 .rnd
-rw-r--r-- 1 movies movies 290 28 Jan 2005 .schroedinger.cfg
drwxr-xr-x 33 root staff 1122 23 Jun 17:27 .spumux
drwx------ 3 movies movies 102 24 Mar 12:06 .ssh
-rw-r--r-- 1 movies movies 85 12 Jan 2005 .sversionrc
-rw------- 1 movies movies 6205 23 Aug 13:28 .viminfo
drwxr-xr-x 8 movies movies 272 22 Feb 2005 ACCOUNTS
drwxrwxr-x 6 movies movies 204 20 Aug 09:37 Backup

The "d" is for directory

The next three tripletes correspond to:

Owner, Group, World
rwx rwx rwx

"r" read
"w" Write
"x" Execute

YOu can set the permissions on anything that can be listed or hidden so long as you know its name and it either belongs to you or the group you belong too

example: One of the directories was called "Backup"

chmod 700 Backup

ls -ald Backup

Will change to:
drwx------ 6 movies movies 204 20 Aug 09:37 Backup

This means only "Owner" can do anything with this directory, except root which has access to anything.

There are other settings which change the effective permissions on an executable/app whilst you're using it "SETUID", a mixture of these that can cause mayhem in a complicated multiuser system with C1 or C2 security.
This discussion on the GideonSoftworks site is the best discussion of the entire topic of ownership and privileges I have seen. You cannot discuss privileges without ownership and vice-versa.
It actually goes even further than that, because there is a fourth triplet, i.e.

man chmod.

Sticky bits, setuid and so forth

Some UNICES used to be able to hide files using this method.