If you mean, to 'knock out the background' and not just square-cut an image, you will need the full version of photoshop which includes the pen tool and accompanying clipping paths.
If you are only working in Photoshop, and the images are only for pasting back into Photoshop, then you may be able to do the tedious method of using the lasso tool to make selections (be sure to save your alpha channels) and the paste your selection onto a layer with transparency.
There is no cropping that is other than four straight sides, square or rectangle. Any other 'cropping' as you call it, is called a clipping path or knocked out selection.
If you mean to trim around an object, leaving everything else transparent, you can do that in Photoshop Elements. You can have multiple layers of objects, each with transparent, or partially transparent, sections, and see what lies beneath. Look for Transparency in the help file or manual.
Best to ask in the Photoshop Elements forum Dog, they will have the
knowledge there. In Photoshop you would use the Polygon selection tool
or the Magic Selection tool, but I don't know if these options are
available in Elements.
You can do it with Photoshop. I don't have Element, can't say.
For Photoshop. Use the Magnetic Lasso tool and trace around the boundary of
what you want to keep to form a close loop, Photoshop auto-detects this
boundary for you, good 99% of the time. Copy it to clipboard. Then, open a
"New" image with transparent background - photoshop automatically create one
that has the same size as the clipboard. The next step is to paste the
clipboard to the new image. done.