Because the button itself has 2 states: Pressed and Unpressed, but in some cases in particular speedruns, that pressed/unpressed state makes a difference whether or not you count a press or not. For those cases a "half" press refers to the idea that the button was pressed before the segment started.
Not on the hardware level, no, it's really not.
it's either being pushed for X number of frames or it isn't.
Pressed is when it's state goes from 0 to 1 and unpressed is when it goes from 1 to 0.
A half press means the state became "pressed" or started at 1 at the time the recording started. Thus the 0 -> 1 change on that button never happened in the run, but the 1 -> 0 change did.
Therefore half the state changes needed to denote a press didn't occur, hence the name, half-press.
In a full run, the button would go from 0 -> 1 in a prior segment before the next one started and is considered a full press, the half press is only used when looking at a sub of a full segment individually.
let's take smash bros for example: if A is 0 the fighter will do nothing. if it's 1 the fighter will start charging the hit. if it's 2 the fighter will stop charging and hit his opponent. then the value will go back to 0