Quick and dirty: the basic permissions are read, write, and execute, and are applied to the owner, the group, and everyone else. They’re applied to all files and directories individually.
It’s represented by a 3 digit number (in octal, which is base 8, so 0 to 7). The first number is the permission given to the file’s owner, the second to the file’s group owner, and the third to everyone else. So, the owner of the file is the one user account that owns it, the group applies to all members of that group. User and group ownership are also applied to each file and directory individually.
Read, write, and execute are represented by the numbers 4, 2, and 1, respectively, and you add them together to get the permission, so 0 would be nothing, 1 would be execute but not read or write, 2 would be write but not read or execute (and yes there are uses for that), 3 would be write and execute but not read, 4 is read only, etc through to 7 which is basically full control.
This will take a little bit to make sense for most people.
chmod (change modifier, I think) is the program you use to set permissions, which you can do explicitly by the number (there are other modes but learn the numbers first), so chmod 777 basically means everyone has full control of the file or directory. Which is bad to do with everything for what I hope are obvious reasons.
chown (change owner) is the program you use to set the owner (and optionally the group) of a file or directory, and chgrp (change group) changes the group only.
It gets deeper with things like setuid bits and sticky bits, and when you get to SELinux it really gets granular and complex, but if you understand the octal 3 digit permissions, you’ll have the basics that will be enough for quite a lot of use cases.
(Additionally to the 3 digit number, permissions can be represented a bit friendlier where it just lists letters and dashes, so 750 (full control user, read and execute group) could be shown as rwxr-x—, where r=read, w=write, and x=execute, and what they’re applied to can be represented by the letters u for user (aka owner), g for group, and o for other)