Is this LED supposed to be on all the time?
Maybe.
It totally depends on is you want it to be or not.
It can even flash if you want.
But without your sketch it is impossible to say.
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If you have an original Uno R3, the led is driven by a comparator opamp. Some noise from the environment in the input of the comparator opamp is picked up and turns the led on.
This might apply to other boards as well, look at the schematic s
If you have another board (clones or e.g. a SparkFun RedBoard) without the comparator opamp, the led will be off because an Arduino pin as input is a high impedance.
It should certainly blink when you press reset (especially if you can still upload to it!)
If the LED is on continuously even if you upload a sketch that does:
pinMode(13,OUTPUT);
digitalWrite(13,LOW);
then that is indicative of hardware damage.
The opamp that the full-size boards (like Uno) uses to buffer pin 13 (so the LED doesn't load down the pin) is one of the easiest to damage if the board is abused electrically, and when that happens, it generally ends up with the LED stuck on. It's rare to see an official board with this symptom that can be uploaded to (or indeed, even detects on USB), as the sort of abuse that trashes the opamp will also usually trash the 16u2. However, clone boards use more durable (if less versatile) adapters like CH340G, so it's more possible to end up in this situation.
Boards with this symptom should be discarded and replaced - the opamp also does power supply switching, so the external power jack and Vin pin probably won't work, and I suspect the blown opamp would interfere with pin 13, and it's also sitting there getting hot and wasting power.