My MacBook won't recognize my external hard drive (Passport - that I use to back up the Mac!)
By "Passport," I am assuming that you are referring to a Western Digital external drive, which you have been using to perform regular backups. I'm guessing that by "backups," you have been using Time Machine.

Considering that this is a new issue, it is fair to say that the external drive has been formatted correctly for use with Mac OSX (i.e., Mac OS Extended (Journaled)). FYI, if you wanted to confirm the File System, then you can use the Disk Utility to display Info about the drive.

The first rule of troubleshooting is to obtain a known working device, which you can substitute to test and narrow down where the problem lay. So in this case, get access to another external hard drive (perhaps from a friend), and see if it can be accessed.

However, you observe physical issues with your MacBook, such as "spinning noise" and it being unresponsive (slow). Unfortunately, this is a classic symptom that the hard drive within the MacBook is failing. Again FYI, if you wish to inspect the processes on your system to potentially identify other issues, then use the Activity Monitor.

Assuming that the hard drive within the MacBook needs to be replaced, the good news is that the Time Machine backup on the Passport can be used to restore your system after the repair.