An oracle (at least in this context)
is simply an operation that has some property that you don't know,
and are trying to find out.
The term "black box" is used equivalently,
to convey the idea that it's just a box that you can't see inside,
and hence you don't know what it's doing.
All you know is that you can supply inputs and receive outputs.
In the circuit diagram you depict, it is just the Uf box.
Everything else is stuff that you are adding in order order
to help interrogate the oracle and discover its properties.
To give the oracle, you can write it in any valid form that defines a map from all possible inputs to outputs.
This could be a matrix (presumably with an unknown parameter),
or it could be the map $U:(x,y)↦(x,y⊕f(x))$ (strictly, $∀x,y∈{0,1}$),
because given either description, you can work out the other.