I am trying to run a script (not created by me) and have some basic questions. Consider this command:

bash update_repo.sh ./delta_repo.tar.gz

When I run this I get:
tee: /opt/repo/update_repod.log no such file or directory
tar (child) ./delta_repo.tar.gz cannot open no such file or directory

So I am thinking something in the path is wrong or where I have the delta_repo.tar.gz is incorrect. I was curious
on what the ./<filename>. Especially what the ./ means.

Thanks.

...ar

In the current directory. You would use that if the current dir is not in your $PATH.

test.sh

Actually the . in ./<filename> is a shortcut for current working directory i.e the output when you execute the command pwd.
The output of pwd does not contain a trailing / and therefore you to to add it to separate the directory from the command/file you are trying to run or access.

linux only searches the environment path but not current working directory (i.e. where you are in the filesystem ) when trying to run a command so . (./) is used as a shortcut to specify the absolute path. On the other hand if a file is used as a command line argument as in your example the script does not need a ./ if the file is in the current working directory as that of the script you are running.

Your particular error "tee: /opt/repo/update_repod.log no such file or directory" maybe due to not having a subdirectory called repo within your /opt. Without knowing anything about this script, what it is supposed to do or delta_repo.tar.gz it is difficult to provide more specific answers.

This command is running the script update_repo.sh with the argument (file name) './delta_repo.tar.gz'

Those errors are occurring because something within that script is encountering an error.

If you want actual help we would need to see the exact content of that script and what it does so the commands and the error could be parsed.

As a side note to the good advice above, running an unknown script with zero knowledge on what it might do would not be a good suggestion IMHO

As mentioned above, maybe attach or display the script here and we can see if we can help you and advise what it does?