Execute but not read permission for a shell script

Basically it means you can tell the operating system to run the code in the file. For example, if the file was a binary executable, read access would allow you to view it, write access would allow you to modify it, but without execute permissions you would not be able to run the program. In the case of a script, its a little more complicted, because you don't necessarily need to 'run' the program, you can just read its contents into an interpreter, which itself has the execute privelige, but you do not need execute permissions on the script itself.

Some scripts in Linux are themselves executable, you will often see a line at the top like

#!/bin/bash

or

#!/bin/python

That line tells the kernel that the file can be executed by calling the relevant program (and isn't just text). Then you can just run your script like

./script

instead of having to do

python ./script

Yes, although execute permission without read permission is meaningless on a non-binary file (since you need to read a script to execute it). On a directory, execute without read means you can traverse the directory, but not list it or do anything else with its contents. Consider the following path:

/home/user/foo/bar

If directory foo has mode 0711 (full permission for owner, execute only for group and world), while directory bar has mode 0755 (full permission for owner, read and execute for everyone else), then accounts other than user may cd /home/user/foo/bar and find that bar behaves as any ordinary directory; however, while cd /home/user/foo will work, ls and any other command in that directory will fail due to insufficient permissions (i.e., you can't read the directory to list its contents).

Write permission without read permission on a file does just what it implies: you can write to the file, but you can't read back what you've written. This might be useful in a case where processes under multiple accounts write to a single logfile, but a process belonging to one user must not be able to read the (presumably sensitive) log entries from a process belonging to another user.