Which are the most effective modern intuitive definitions of a vector? First, I would like to clarify what I mean by "intuitive definition": an intuitive definition is an informal understanding of a concept which helps to build mental agility, with the possible side-effect of crippling one's ability to imagine the full scope of the concept.
Next, I would like to clarify what I mean by "effective intuitive definitions": effective intuitive definitions are sets of intuitive definitions, which, taken together, try to minimize the overall crippling effect of intuitive understandings on imagination.

In this document, in section 1.1 the author writes:

I’d say the best intuitive deﬁnition of a vector is “anything that can be represented by arrows that add head-to-tail.”

I am not convinced that is the best intuitive definition. I think that at the very least, for any concept, the best intuitive definition must be a set of intuitive definitions that is an effective intuitive definition.
In particular I feel that the above proposed "best intuitive definition of a vector" is severely cripples the imagination because it ties down one's imagination to three dimensional space.
What are some effective intuitive definitions of vectors?
 A: To me, vectors are objects I can add together (or subtract) as well as scale up or down by a constant. This is essentially the definition of a vector space, that is, an abelian group with an action of the units of a field. This allows you to clearly see why $\Bbb R^n$ is a vector space just as well as spaces of real-valued functions, modular forms, etc.
A: In geometry, a vector is the mathematical entity that encodes a particular translation in space. Such a translation has a "direction" and an "amount" $\ell>0$; therefore it can be envisaged as an "arrow" of length $\ell$ drawn anywhere in the figure. At the same time such a translation affects the coordinates of all points in the same way: it adds "componentwise" a certain triple $(a_1,a_2,a_3)$ to them. The "algebraic view" is then that the vector is tantamount to this triple.
A: I think it is importation to make a distinction between a vector in an arbitrary vector space and a vector in an arbitrary inner product space.
My intuition for vectors in a vector space is just "things you can add together and scale", just as in the answer by Dylan Yott. I know, in the back of my mind, that I can pick a basis and express everything in terms of this basis, but I'll resort to that only as a technical tool.
My intuition for vectors in an inner product space is more "arrows, with a length and direction" and I tend to think of the space as having some "axis" (basis). I remind myself that I can still "move the coordinate axes around" (change basis). So, I typically have ${\mathbb R}^n$ in mind.
That said, my experience is mostly in abstract algebra, where the first notion (vector space, or more generally, free module) crops up much more frequently than the second notion (inner product space). That may skew my intuition.
A: A real $n$-dimensional vector $v$  is a map $v:\{1,\dots,n\}\to\boldsymbol R$. This map may be considered as a translation and thus represented by an arrow.
A: To quote Vector from "despicable me": I'm applying for a villain loan. I go by Vector. It's a mathematical term, represented by an arrow with both direction and magnitude. Vector! That's me, because I commit crimes with both direction and magnitude. Oh yeah! 
Examples of vector quantities from real life, illustrating Vector's words (without the crime), arise naturally in physics. Certain forces have direction and magnitude, and the way such forces act can be seen by applying vector algebra. Changes to the force can often be seen as scaling or changing the direction. 
I don't know if this is the most effective intuitive definition. I find it effective and, at the very least, amusing. 
