What's fractional share trading? Here's how it works, and who's offering tiny slices of pricey stocks like Amazon and Shopify.

A wave of brokerage firms and fintechs have started allowing customers to trade small slices of individual stocks.  Many are marketing the offering as a way to democratize investing and allow average customers to own popular but highly priced stocks.

It's all part of a fierce competition for young investors, especially high...