Brokers offer fractions by buying whole shares and allocating slices across customers on their own books. The fraction exists only at that broker: it cannot be transferred to another firm, cannot be moved to direct-registration, and is normally liquidated for cash when you transfer an account.
Fractions also arise naturally from a drip, from a stock-split with an awkward ratio, and from a spin-off. In corporate actions fractions are usually cashed out rather than issued, which creates a small taxable event.
Example: $500 into a $327.40 stock buys 1.527 shares. On transferring the account, the 0.527 fraction is sold at whatever the price is that day and $172.60-ish in cash moves instead.
Related: drip, stock-split, round-lot, direct-registration, cost-basis