The standard player is Raspberry Pi based, and its LEDs will often tell you what is wrong before the screen does. This covers the Raspberry Pi 3 B+, and the same lights appear on nearby generations.
A Player Max has no LEDs of this kind. It has a power button with an integrated status indicator instead.
Power and activity
Two LEDs sit next to the power connector.
The red LED, labelled PWR on the board, is the power supply. A steady red light means power is good. Flashing, blinking, or off means the supply is not delivering enough power, which is the single most common cause of a player that reboots or freezes at random.
The green LED, labelled ACT, is storage activity. Irregular flashes are normal and happen whenever the player reads or writes its card. A steady green light while booting means a missing or bad microSD card.
Ethernet
Two LEDs sit on the ethernet socket. Both are off when the player is using Wi-Fi, which is expected rather than a fault.
The yellow LED on the left is link speed: on for 100 Mbps, off for 10 Mbps.
The green LED on the right is the link itself: on when a link is established, flashing during activity, off when there is no link.
Reading them together
Steady red and no green activity on boot points at the card. See Flashing problems.
Flashing red points at the power supply, whatever else the screen is doing. See Build your own player for why the official supply matters.