Fractional shares are pieces of a stock less than one whole. These can result from various actions such as stock splits or dividend reinvestment plans (DRIP). When a stock doesn't split evenly, investors owning an odd number of the shares will receive portions of the share. If you owned 5 shares and there was a 5:2 split, you will end up with 12.5 shares. The 0.5 share would be considered the fractional share. With DRIPs, you are putting the dividend received from the share as an investment back into the company. Since dividend values are generally less than the share value, you will receive a percentage of the share rather than a full share. If you have a single stock valued at $100 and receive a $1 dividend, that $1 would get invested into the stock resulting in you owning 1.01 shares of the stock.
It is also possible to simply buy fractional shares through some brokerages such as Robinhood. For investors with a limited investment fund who can't afford to buy high priced stocks, this is a great option for diversification and growth. If you only had $100 to invest and wanted to buy Amazon (which is around $3200 at time of writing), you can buy a .03125 fractional share (100/3200). You can even distribute the money across multiple stocks by buying a smaller fractional share (like .01 for $32) and investing the rest elsewhere. Since the purpose of stock splits is to attract more investors with a lower stock price, being able to buy fractional shares actually makes stock splits less useful.
Fractional shares also allow you to fully invest your money. If you had $100 to invest on a stock valued at $30, you'd buy 3 shares and have $10 remaining. That remainder will be sitting as cash until you have $20 more to be able to afford another share. With fractional share, rather than having to wait, you can invest soon and just buy 3.33 shares right away.
Overall, fractional shares are a great way to get involved in investing. The only caveat is to make sure the brokerage you're using supports fractional shares since not all do. This will impact trading fractional shares as well as receiving fractional shares in an odd stock split.