malfunction of any of the various burner components (pump, fan, ignition system, primary control, burner motor nozzle)
electrical power to the water heater is off
clog in the fuel line
leak from the fuel line
empty oil tank
The implication is no hot water.
Flow hot water at a faucet to test the operation of the water heater. The oil burner should ignite within a few minutes.
If the burner does not ignite immediately, check the electric power switch. In most homes with basements, the switch is near the bottom of the basement stairs. Most codes require the switch to be between the stairs and the appliance. The switch may be a conventional light switch or a fused disconnect. If the switch is on, check the fuses or breaker panel.
If there is electricity, but there is still no response from burner, check whether there is oil in the tank.
If the tank is not empty, the problem is likely in the fuel line or at one of the burner components.
If you are technically inclined, you can check the refractory for pooling oil. The presence of oil may indicate the pump is working but the fan or igniter is not. Do not reset the controller if there is a puddle of oil in the refractory, as an explosion can result. If you can't see inside, don't reset the controller. The nozzle may be clogged. You can try the reset button on the primary controller once to attempt ignition again,
if you are sure there is no oil puddle