Reference: chmod syntax

tl;dr: [Special,] User, Group, Other. 4 (Read), 2 (Write), 1 (eXecute)

Entity Operator Permission. [ugoa]*?[+-=][rwx]*

chmod is a command used to change the permissions of a file or directory in unix-like systems. It manipulates the permissions number of a file or directory, which acts as a bit mask for each individual permission.

Bit Mask: As all numbers can be represented in binary, a bit mask is a number that compacts multiple boolean values into a single number. For example, the number 5 in binary is 101, which can be used to represent three boolean values: true, false, true.

In its minimal form, the permissions number is a 9-length binary number, where each 3 bits represent the permissions for a specific entity: the user, the group, and others. The first bit in each group is the read permission, the second is the write permission, and the third is the execute permission.

As an example, this is what read-write permissions for the user, read-only permissions for the group and others, would look like in binary:

      U  G  O
     ╭─╮╭─╮╭─╮
     110100100
     │││││││││
User RWX││││││
Group   RWX│││
Other      RWX

To make this easier to read, most outputs will replace 0 and 1 with - and the permission letter, respectively. For example, the above example would be displayed as rw-r--r--.

Since binary is kind of verbose and hard to read, chmod accepts the permissions number in octal form (0-7), with a digit for each entity. In simple terms, the read permission adds 4, the write permission adds 2, and the execute permission adds 1. For example, read-write permissions for the user, read-only permissions for the group and others, would be represented as 644.

Here's a small tool to help you convert between the two:

In addition to directly setting the entire permissions number, chmod also has a syntax to modify the permissions of a file or directory. This takes the form of an optional set of entities, an operation to perform, and the permissions to modify.

• The entities can be one of u (user), g (group), o (others), or a (all). Multiple entities can be combined by listing them together, for example ug or go. If no entities are specified, a is used (mostly; see note).
• The operation can be one of + (add), - (remove), or = (set).
• The permissions can be one of r (read), w (write), or x (execute). As with the entities, multiple permissions can be combined by listing them together, for example rw or wx.

NOTE: Specifying no entity is nearly the same as specifying a, but the umask of the current shell is unchanged. The umask is the permissions number that is removed from newly created files by default, and usually doesn't need to be changed.

Here's a small tool to generate and parse the syntax: