The piece of skin that your mother is referring to is called the "hymeneal ring". All women with some rare exceptions are born with an opening in their hymen. The opening is usually big enough to allow vaginal discharge and blood to pass through. The opening can become enlarged with physical and sexual activity. The natural opening should be large enough to allow for tampon insertion. (see Pain Inserting Fingers During Masturbation). If your hymeneal ring is still intact, this could be causing the tampon to feel "stuck".
Another reason for the tampon to get stuck is nervousness. Nervousness and stress cause the muscles around the vaginal area to tighten. The remedy is to relax. Try inserting a tampon at a time when you are not rushed. Use a regular size (some brands have a junior size) with an applicator - most people find them easier to handle. You may find using a plastic applicator easier than a cardboard one. Take a few deep breaths and concentrate on relaxing your vaginal muscles. Following the directions on the package - put one foot on the toilet and put the tampon in while it is aimed toward the small of your back, sort of a 45-degree angle. Some people squat and insert the tampon while it is aimed almost all the way toward your back. Using lubricant, like KY Jelly on the tampon may help as well.
Remember to change your tampon often to avoid toxic shock syndrome (TSS). If you continue to have difficulty, a physician like an adolescent medicine specialist or a gynecologist may be able to guide you.