The file I'm trying to run is in the root folder. Do I have to sudo to the root in order to change permissions? I am currently logged in as an admin.

no.
files in root's domain should have root permissions. don't change it.

unless you provide much more detail there's nothing more to say.

What do you mean by "logged in as an admin"? In Linux root is the only superuser. Other users generally are restricted to their home folders until they run sudo while being in the sudo group.

If you run "sudo command" you run the command as root, no matter the command is run from a restricted user's session. That was your question I guess?

There are three permissions, read, write and execute that are assigned to a file. Their are three sets of permissions, owner, group and user. Only the owner of the file can change its permissions, except root which can change any permissions.

I encountered a problem while trying to upload a Arduino sketch to a ESP8266-01S module on my mac OS 10.11.6 (El Capitan). The Arduino/Esp community said the problem was with a file named"esptool" and suggested replacing that file with one named, "esptool.py". Following the instructions for installing esptool.py, the file was installed in the following path:

Macintosh HD/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/esptool.py

The next step was to modify a file named, "platform.txt" with the path indicated above. However, when I tried to upload the sketch, I got this error message:

java.io.IOException: Cannot run program "/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/esptool.py": error=13, Permission denied
at java.lang.ProcessBuilder.start(ProcessBuilder.java:1048)
at java.lang.Runtime.exec(Runtime.java:620)
at java.lang.Runtime.exec(Runtime.java:485)
at processing.app.helpers.ProcessUtils.exec(ProcessUtils.java:11)
at cc.arduino.packages.Uploader.executeUploadCommand(Uploader.java:129)
at cc.arduino.packages.uploaders.SerialUploader.uploadUsingPreferences(SerialUploader.java:207)
at cc.arduino.UploaderUtils.upload(UploaderUtils.java:78)
at processing.app.SketchController.upload(SketchController.java:713)
at processing.app.SketchController.exportApplet(SketchController.java:686)
at processing.app.Editor$DefaultExportHandler.run(Editor.java:2135)
at java.lang.Thread.run(Thread.java:745)
Caused by: java.io.IOException: error=13, Permission denied
at java.lang.UNIXProcess.forkAndExec(Native Method)
at java.lang.UNIXProcess.<init>(UNIXProcess.java:247)
at java.lang.ProcessImpl.start(ProcessImpl.java:134)
at java.lang.ProcessBuilder.start(ProcessBuilder.java:1029)
... 10 more

Now I need to know what permissions need to be changed, and how to change them.

In the "Get Info" window of the Mac OS Funder, I was able to change the permissions for all files in the path except for the first 2 folders. Apparently, I lack permission to make changes to those 2 folders.

Still getting same error message.

Check the perms on esptool.py. I'm not a Mac expert, but I think if you right click on the file there should a drop down with permissions on it.