It's hard to make sense out of this question. But suppose that the action at the nut is too low (the nut is slotted too deeply for that string or whatever). Suppose it is so low that the string basically "self frets" at the first fret. That explains how there is no difference when fretting at the first fret versus not fretting: the same pitch is produced.
In that case, if we tune the string to E, we are actually tuning what should be the F note to E; the string itself is actually half a step down. Then if we try to play the second fret that should be F#, we get F. That explains that the second fret produces the note that the first fret should produce.
If the nut action is too low on an acoustic guitar, there are various solutions. If the original nut is re-used, it has to be shimmed to make it higher. This means removing the nut, and inserting a shim (such as a strip of wood veneer) between it and the neck. Then if the other string actions become too high, their corresponding nuts in the slot have to be filed to make them a little bit deeper.
As a quick and dirty temporary hack just to get playing, you could stick something between the nut and the string to prop the string up a little higher, such as a small piece of pop-can aluminum bent into a tiny half-pipe shape.
It's possible that your guitar was designed for low action near the nut, and someone has over-tightened the truss rod, creating a backbow. Before doing anything with the nut, I would check how straight is the neck. If there is a backbow due to the truss rod being tight, it should correct itself by loosening the truss rod. (If there is a backbow in spite of the truss rod being loose, that is bad news: the neck is actually warped.)
Lastly, perhaps that first fret itself is not inserted properly into its slot. We say that the "fret is out", in guitar jargon. Even if frets are seated into a neck properly, they still have to be leveled (basically sanded to the same height) and crowned and dressed (reshaped and polished). This is not always done in the factory for mass produced, inexpensive guitars. They just stick the frets in and ship it.