So I have a question regarding these such weapons and items in the game. I am not exactly sure what they mean by leveled.

Say I get a certain item at the early level say mid teens, would it continue to increase in damage as I level up into the higher levels?

Once you acquire an armor or a weapon, its properties never change (with the exception of the Ebony Blade, but that's a sub-quest on its own).

Leveled quest rewards are scaled in power based on the player's level at the time the reward is given. They don't improve when you level up; once you get it, that's it.
For example, Nightingale Blade is enchanted with health and stamina absorption: 5 pts if you acquire it on levels 1-18, 10 pts on level 19-26 and so on.

http://www.uesp.net/wiki/Skyrim:Leveled_Items a list of such items.

No. The stats the item has when you receive it will stay the same even when you level up.
Here's a link to a list to all of the leveled items in the game http://www.uesp.net/wiki/Skyrim:Leveled_Items
So I would suggest getting these items when you are the optimal level.

Ahh I see. That is what I wanted to know. Cause I was hoping to get the Nightingale Bow and Blade early on. But its best value is around level 46. So it doesn't really make sense completing the quest in the mid teens.

If you're on PC, you can use http://skyrim.nexusmods.com/mods/2604 and never feel punished for doing a quest too early again.