Hi there, & Welcome to Tom's Hardware!
Couple of things to check on initially. Make sure your removable HDD is powered up, light is on, and you can hear the HDD spinning (ear to the drive)
Then try a different USB cable and connect to a different USB port.
Connect when your computer is off, then boot to the BIOS and in the BIOS, see if the removable drive is recognized there on the CMOS page usually.
If so, then completely booted up, go to Disk Management, and in the lower graphical section, report what is listed in the Drive Status column (Disk 1 or 2, Basic, size, Online) or similar, and in the Volume Status column (VolumeLabel, DiskLetter, size, Healthy (Primary partition) or something similar. Also what is the color of the band above the Volume Status area. Should be dark blue.
Report back and we can figure out what the problem is.