Symbolic-Mode
[ugoa] [+-=][rwx Xst]
ugoa specifies which users' access to the file will be changed:
  user who owns it
  Other users in the file's group
  other users i.e. not the user who owns it and not users in the file's group
  all users ( i.e. ugo)
Default is a but bits that are set in the umask are not affected.

+ adds to the existing modes ,
- removes modes, and
= sets modes

rwx Xst select the modes for the specified users:

read , write, execute
eXecute only if the file is a directory or already has execute permission for some user
Set user or group ID on execution^†
Save program text on swap device ( a performance enhancer )

Examples

Deny write permission to all: chmod a-w file
Allow read permission to all: chmod a+r file
Make a file readable and writable by the group : chmod g+rw file
Allow everyone to read and execute the file and turn on the set group-ID:
chmod =rx,g+s file

Numeric mode:

The mode is calculated by adding the values
Owner (rwx) = 400+200+100= 700 ; Group(rx) = 40+10 = 50; Other (rx) = 4+1 = 5
mode = 755
Setting mode to 777 should be avoided as this permits anyone to modify the file.

• Allow owner read+write+execute, users in the group read+execute, everyone else noaccess
  chmod 750 file
ls -l file
-rwxr-x--- 1 dgerman users 978 22 May 19:10 file
  
• owner read,write,execute , group read+write , others no access
  chmod 760 file
ls -l file
-rwxrw---- 1 dgerman users 978 22 May 19:10 file
  

The mode of symbolic links are not changed since they are not used.
The mode of the target file are changed except during recursive directory traversals.

This page documents the GNU version of chmod.

see also:
ls -l lists current permissions for -- u (owner) rwx -- g (group) rwx -- O (Other) rwx --
chgrp - Change group ownership
chown - Change file owner and group
umask - sets defaults ( actually the complement !
Windows equivalent commands:
ATTRIB - Change file attributes
CACLS - Change AccessControlList settings
XCACLS - Change file permissions

and for us Mac ers