How does the octal (absolute) notation of chmod work, and what are chmod 644 and chmod 600?

Each right has a value (r = 4, w = 2, x = 1); add them per class to get one digit each for user, group and others.

* Four digits: special bits, then one digit per class, each the sum of r4, w2 and x1. *

• chmod 644 file gives rw-r--r--: the owner reads and writes, everyone else reads.
• chmod 600 file gives rw-------: only the owner can do anything. This is the right setting for something like a file of financial data or an SSH private key.
• chmod 755 dir gives rwxr-xr-x, the normal mode of a new directory.

The full form has four digits. The optional first one holds the special bits: SUID = 4, SGID = 2, sticky = 1. So chmod 1777 /tmp gives rwxrwxrwt, and chmod 4755 sets SUID on a program.

Worked example: a directory is rwxr-xr-x (755), and others should get write as well. Others become rwx = 7, so the command is chmod 757.

Go deeper:

• Wikipedia: chmod — octal and symbolic modes with many examples.
• man7.org: chmod(1) — the reference, including the special-bit digit.