Is a tensor of order 0 an invariant or a variant? Is a tensor of order zero invariant or a variant?  
If I understand correctly: 


*

*a variant implies that in different coordinate systems it has a different value.  In contrast

*an invariant implies that the value doesn't change between coordinate systems
Also, is length of a vector considered a tensor of order zero or is that something different? what would be an example of a tensor of order zero?
 A: Tensors of order zero are invariant, and tensors of order greater than zero are variant.
For example, if we contract two variants, a contravariant tensor $W^i$ with a covariant tensor $\vec{V}_i$, we contract away all of the indexes, and we get an invariant U:
$$W^i\vec{V}_i = U$$
In barred coordinates:
$$\bar{W}^i\bar{\vec{V}}_i = \bar{U}$$
To Rephrase, since tensors of order greater than zero are variant, it follows that the components between barred and unbarred coordinates will be different values ($W^i$ vs. $\bar{W}^i)$, and bases vectors between barred and unbarred coordinates will be different ($\vec{V}^i$ vs. $\bar{\vec{V}}^i)$, however the product of the component and the bases vectors with produce an invariant that is the same regardless of coordinates system:
$W^i\vec{V}_i = U = \bar{U} = \bar{W}^i\bar{V}_i $
Length of a vector is an example of an invariant. In fact, all invariants are tensors of order zero.   They are invariant because they have no indexes, and consequently they don't need a Jacobian to transform them between coordinate systems, thus they don't transform to another value when coordinate systems are switched.
However, not all non-indexed objects are invariant... for example the determinant of the metric tensor is a variant even though the result of taking its determinant has no indexes.  Only non-indexed objects that are also invariant between coordinate systems changes are tensors of order zero.
