A gem, or jewel, is a cut and polished precious or semi-precious stone that can be used to make jewelry. A gem can also be used to describe something that stands out as being better than everything around it, as in "this figurine is the gem of her collection".
A hem is the stitching along the edge of a garment. Hems make the edges of clothes look nicer, and stop them from fraying.
Jerusalem is the capital of Israel. It has a population of around 1 million.
A modem is a device that transmits data over telephone or cable lines. You connect to the internet using a modem.
A problem is any question or situation that is difficult or complicated. The cat in the picture has a problem. It is stuck up a tree and doesn't know how to get down.
Salem is the capital of the state of Oregon in the United States of America. It has a population of around 155,000.