Hello there,
Comparators are often viewed as three terminal elements with a power supply.
The two inputs are "compared" (hence the name, "comparator") and if the voltage on the non inverting (+) terminal is greater than the voltage on the inverting (-) terminal the output goes to a high state, otherwise the output goes to a low state.
The two output states, high or low, are defined slightly different for different part numbers. Many comparators have an "open collector" or "open drain" output, which simply means that the output goes to a high impedance state rather than to a high voltage level. This means you may need a pullup resistor to get the output to actually produce a high output voltage. Most comparator output voltage ratings are limited though, so you can usually only pull up to the supply voltage but no higher.
When driving an LED, it usually makes sense to connect the LED to the output so that it turns on when the output goes low. The cathode is connected to the output, the anode is connected to a resistor to limit current, and the resistor is connected to the most positive voltage supply like +Vcc.
Of course you have to have enough voltage to drive the LED though, so if you are using a white or blue LED for example you should have at least around a 4v supply voltage available. Otherwise you should use a red LED because red LEDs have lower voltage requirements, which means many of them can turn on with as little as 1.8vdc. The current limiting resistor has to be chosen to cover the full range of voltage however so that the LED always has a reasonable current level. If this is not possible, then a constant current circuit can be used to drive the LED.