There are several possible answers here.
In Finder, click on
Finder at the top of the screen, then
Show these items on the desktop make sure
External Disks is checked.
Similarly, click on the
Sidebar tab, and tick External disks there.
Now close the Preferences window and check your Desktop and Finder sidebar.
If you still don't see anything, make sure Finder in selected, then click on
Go to folder. In the window that's dropped down, type in
/Volumes/ and hit Enter. You should now be in a Finder folder called Volumes that contains links to all of your connected drives and mounted disk images. Is your USB drive there?
If not, let's see if it's connected but not mounting. Click on Spotlight, and type in
Disk Utility. It might pop up before you finish typing. When you see it, hit Enter and let it load.
In Disk Utility, you should have a window with a sidebar inside it. In the sidebar, you should see your internal hard drive and any external hard drives you have connected. Can you see your USB drive in there? If so, but it's greyed out, click on it and click the big blue
Mount button. Hopefully the hard drive should start working, and then appear on your Desktop.
If it refuses to mount for some reason, or you don't see it there at all, then it's unfortunately likely that there's something a bit wrong going on, either with your connection or the drive's hardware. That's beyond the scope of this answer, but I'll keep my finger's crossed that it doesn't come to that.