books say that PATH variable contains the locations where the shell searches the commands to execute them.

I created a shell script and saved it in /home/user/Downloads directory and then i ran this script from the /home/user/Downloads directory. It did not execute.

however when i did set the path
export PATH=$PATH:/bin/sh within /home/user/Downloads directory, the shell script executed. Does it not mean that PATH variable is holding the locations of the shell program rather than the commands that are to be executed?

everytime i have to give the location of the shell i.e /bin/sh in the PATH variable no matter from where i am executing my script

That is correct.
Right...so when you set the path, the system knows where to look for commands. Nothing wrong there. So either move your script to a directory that's already in your path, or modify it to include whatever directory you want. Your "books" should also have told you this, too. Modify your path by editing your .bashrc/.profile/.bash_profile file (whichever applies), and each time you launch a shell, that PATH will be set.

Also, this goes well with your other homework-sounding question, and given that you've been using Linux for four years now, seems a bit odd.

Wrong. We only pass directories to the PATH variable and not location of executable like /bin/sh.

Please re-read what was posted. The OP said they had to give a location to the shell, FOR EXAMPLE, /bin/sh. Since /bin/sh IS the fully-qualified path to sh, there is NOTHING wrong with that. All that says, is that /bin is NOT in the OP's path.

Please don't post incorrect or misleading information.

See what he posted. That's obviously wrong.

So in your case, if your script is located in /home/user/Downloads you need to add this directory into PATH: PATH=$PATH:/home/user/Downloads in order to look for your script to execute it (because you want to execute your script, not /bin/sh).

READ and UNDERSTAND what he posted before saying that. You *CAN* do exactly what he said, and gave that only as an example.

Putting the literal path to an executable into PATH doesn't work. PATH should hold directories:

The OP may or may not have meant that (perhaps it's ambiguous, though I do read it the same way as veerain does).

the situation still persists.

When i have given the path as /home/user/Downloads and when i try to execute from some other location, it says that file does not exists. But did I not put the path in the first place. Then why am I getting such a message

You need to back up and describe what, EXACTLY, you're setting the PATH to, and what, EXACTLY, you're trying to run. Don't just say "I'm doing X", you need to copy and paste the actual terminal output, both your commands and the result, in order for anybody to help you.