This is the image of an apple cut into half.
There are two apple halves, or two halves of an apple. (They mean exactly the same)
We can count the number of apple pieces.
A person might ask for half an apple.
This is a picture of some apple slices.
It's not easy to count all the slices, so it's just quicker to say some.
You could also ask for some apple if you wanted more than two slices or pieces.
This is a picture of a single apple and a quarter slice.
The apple has not been cut into four quarters. (Alternatively) It has not been quartered.
Although we can see it is countable, some people might ask for some apple, or ask for the larger/smaller piece.