For much help, you'll need to provide a lot more information. But based on guesses without that, you are not matching your input ("project preset") settings to the right choice of output render choices. If you can list everything from the source of the video to computer and software versions you should get some skilled volunteer help.
What version of Premiere Elements and on what computer operating system is it running?
YouTube uses a 16:9 player. Problems that your video can get into aspect ratio wise can be found at the bottom of the the following YouTube article.
Also, from the point of view of clarity, there is a YouTube settings control that YouTube sets automatically (usually starting at a lower level than your upload)....have you changed that set to HD? And, did you upload interlaced or progressive video from Premiere Elements?
Immediate Situation - Clarify of Upload and Aspect Ratio of Upload
The specific details, besides Premiere Elements version/computer operating system, that would point to the solution would include:
a. Premiere Elements project preset (if you do not know, please tell us what you see for Editing Mode, Timebase, Frame Size, and Pixel Aspect Ratio (even if they look grayed out) under Edit Menu/Project Settings/General.
b. Properties of your source on the Timeline.
Interlaced or Progressive
Pixel Aspect Ratio
Frame Size and Interlaced or Progressive properties would be my starters.
c. What did you use for export settings...you appear to have used
Publish+Share or Share depending on what version of the Premiere Elements you have
What did you set for Presets? And, after you selected the preset, did you use the Adobe default export settings? Or, did you customize them under the Advanced Button/Video Tab, Audio Tab? Why did you not use one of the settings for AVCHD.mp4 (Publish+Share/Computer/AVCHD with preset MP4 H.264 1920 x 1080p or YouTube Widescreen HD)?
d. You apparently uploaded your Premiere Elements file to YouTube at the YouTube web site. Did you try the Premiere Elements built in YouTube feature for the upload? Did you not use it due to file's duration or size?
Answers to the above should expedite your reply and our response with a solution since you will know specifically the details that we seek for us to help you.
Please review and consider. Any questions or need clarification, please do not hesitate to ask.
Add On...after original post...forgot to include YouTube Advanced Settings link...it now appears after this Add On...
Also, an MPEG is not a good format to export if you're going to post your videos to YouTube!
Bill's advice is spot on. The issue is most likely your workflow -- what you're putting in, how you've got your program set up and what format you're exporting. Though, as he says, he'll need to know more about your project (including which version of the program you're running and what model of camcorder your original video is coming from) before he'll be able to give you specific advice.