What the OUI tells you
A MAC address is six bytes. The first three (the OUI, or Organizationally Unique Identifier) are assigned by the IEEE to the device's manufacturer; the last three are chosen by that manufacturer per device. So from the first half of a MAC you can usually tell who made the network chip - useful when an unfamiliar device shows up in your router's client list.
When the vendor is unknown
Two things can hide the real maker. Modern phones and laptops use randomized (locally administered) MAC addresses for privacy when scanning or joining Wi-Fi - these don't map to a real vendor. And some prefixes simply aren't registered. This tool flags the locally-administered bit so you can tell a randomized address apart from a genuine one.
Looking at your router's device list? Pair this with the IP address checker to make sense of the addresses each device is using.