To downsize is to make something smaller. You might downsize your collection of vintage china by selling everything that's nicked or chipped.
You can downsize the items you own by giving them away, and you can downsize your living situation by moving from a huge house to a very small apartment. When a company downsizes, it almost always means it's reducing the number of employees, making the company smaller by laying them off. This last meaning has only been around since the 1980s.