Q:

What is the difference between a zero operator, zero function, zero scalar, and zero vector?

I'm pretty sure that a zero vector is just a vector of length zero with direction, zero scalar is just the number zero, and that a zero function is any function that maps to zero. Not entirely sure what exactly a zero operator is however.

A:

The zero vector is a vector, i.e. a member of whatever vector space is under consideration.  It has the property that adding it to any vector $\bf v$ in the vector space leaves $\bf v$ unchanged.
The zero scalar is a scalar, i.e. a member of the field that is part of the definition of the vector space (usually the real or complex numbers in an elementary linear algebra course).  It has the property that multiplying 
any vector $\bf v$ by it gives the zero vector of the second vector space.
The zero operator is a linear operator, i.e. a linear map from a vector space to a vector space (possibly the same one).  It has the property that it maps any member of the first vector space to the zero vector in the second vector space.
The zero functional is a linear functional, i.e. a linear map from a vector space to the scalars.  It has the property that it maps any member of the vector space to the zero scalar.

A:

In an algebraic context where there is a notion of addition, $0$ is the element such that
$$
x + 0 = x
$$
for every $x$.
If the context is the real numbers, then $0$ is just a number. If the context is the Euclidean coordinate plane, $0$ is the vector $(0,0)$. If the context is the set of real valued functions on the unit interval then $0$ is the function whose value at every point is $0$. If the context is the set of linear operators from one vector space to another then $0$ is the operator whose value at every point of the domain is the $0$ vector in the codomain.
So the meaning of the symbol "$0$" changes depending on the context. That's potentially confusing (which is why you are asking the question.) The advantage in using the same symbol in these different contexts is that it's easy to associate that symbol with its behavior: it's the additive identity.

