Reading the three digits
Each digit is owner, group and everyone else, in that order. Within a digit, read 4 for read, 2 for write and 1 for execute, added together. So 6 is read plus write, 7 is all three, and 5 is read plus execute.
644 is the ordinary file: the owner can edit it, everyone else can read it. 755 adds execute, which is what a directory needs to be entered at all — a directory with 644 can be listed and not opened.
The modes to be careful with
777 means anyone on the system can modify or replace the file. It is the reflexive fix for a permissions error and almost never the right one; the actual problem is usually ownership, which chmod does not touch.
Private keys and credentials need 600 — SSH refuses to use a key that others can read, which is a feature rather than an obstacle.
Frequently asked questions
What does 755 mean?
Owner may read, write and execute; group and others may read and execute. It is the usual mode for directories and for programs.
Is 777 ever right?
Almost never. It lets anyone on the system replace the file. A permissions error is usually about ownership, which chmod does not change.
Why does SSH refuse my key?
Because others can read it. Set it to 600 — owner only. SSH treats a readable private key as compromised.
Something wrong with this tool, or an idea for it? Tell us