Question: It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do better by defining a tensor as a multilinear map from $ V\times...\times V\times V^\ast\times ...\times V^\ast\to \mathbb{F} $, where $V$ is a vector space over $\mathbb{F}$ (often taken to be a tangent space). The transformation law then follows.

My current understanding of spinors feels like the first, dissatisfying definition: they're just "things that transform like spinors" - that is, they're elements of a vector space which transform according to a projective representation of $SO(n)$ which is genuinely multi-valued (i.e. it's not just a true representation of $SO(n)$). We could call this the "spinor transformation law". Note that this is something we've put in "by hand": the way a spinor transforms is not a property of some underlying object, but is built into our definition.

My question is: can we define spinors without reference to the way they transform, just as we did for tensors? Is there some object "underlying" the definition of spinors in terms of transformations, just as tensors are "really" multilinear maps?

\begin{align}
\text{Tensor Transformation Law}&\to \text{Tensors as multilinear maps}\\
\text{Spinor Transformation Law}&\to \text{??? }
\end{align}

Answer: 
I think you're asking for intuition in the wrong direction here.

Suppose that somebody is already familiar with vectors, and wants to understand tensors. That's possible, because tensor representations are built out of vectors, i.e. the rank $2$ tensor representation is just the product of two vector representations, or equivalently a rank $2$ tensor is a bilinear map on two vectors. 

But it's precisely the opposite with spinors. Spinors aren't built out of vectors, instead vectors are built out of spinors! Spinors are the *simplest* possible representations of the Lorentz group, and the vector representation is the product of a left-handed spinor and a right-handed spinor (or equivalently, a vector is a bilinear map on two spinors). 

In other words, asking what underlies spinors is the wrong question. Spinors *are* the structure underlying everything you already knew. You need to rebuild your understanding with the spinors at the bottom, not at the top. 

This [happens a lot](https://physics.stackexchange.com/questions/525262/why-cant-the-schr%c3%b6dinger-equation-be-derived/525265#525265) in physics: you can't ask for an intuitive derivation of a fundamental thing from a composite thing. What you're asking for is analogous to asking which atoms a proton is made of, or how many protons are inside a quark, or how to build a vector out of tensors. (Incidentally, learning fancier mathematics, as suggested in the other answers, *never* answers such questions, because these questions inherently don't have answers. What really happens is that in the process of learning the math, you get familiar with these new elementary objects. Once you can work with them fluently, you stop worrying about explaining them in terms of things you knew before, because you understand them on their own terms.)
Well, you should look at (irreducible) representations of the Lorentz group.
Basically you want all of your ingredients to have correct and consistent transformations under the Lorentz group. 

**The Weyl and Dirac spinors are the most basic objects satisfying that requirement.**

Starting from those you can build vectors as a (multiplicative) combos of two spinors. That's why in old texts you sometimes see spinors referred to as 'half-vectors'. Also, in this context, they use only 'half' of the transformation of a vector, i.e. one-sided vs two-sided.

In that sense its Spinors->Vectors->Tensors.

If you feel fancy, you can also look at things in the context of Geometric Algebra or Spacetime Algebra going back to David Hestenes. Here you can have spinors free of any matrix representation.

Two other references with different perspectives also come to mind: Spinors and space-time (Penrose) and GRAVITATION (Misner Thorne Wheeler)

**The common theme of all approaches is however the special, fundamental transformation properties they have.** You can't get around that.