Are forces vectors? I've learned that forces are vectors... but... in what sense are they vectors?
For example, when I have two forces that are acting on different parts of the body, they produce different results. When calculating how that body will move, I have to take into consideration not only the direction of the force but also the position of which the force is being acted upon. 
I've noticed that textbooks usually assume that the tail of the force vector is where the force is being acted upon, but... if the tail of the 'vector' (force) is important and moving the force vector while keeping it parallel changes the outcome, then doesn't this mean that force is something more than a vector since it can't be moved around? 
In summary: In what sense is force a 'vector'? 
 A: A usual force in classical mechanics is a vector in $\mathbb{R}^3$. This is an object which has a magnitude (strength) and direction. 
If you throw a ball, the force of your hand accelerates it. The speed of the ball depends on the strength of course, try throwing up with different strenghts, and it depends on the direction. You have choices in three dimensions for that, as you can easily test with a ball yourself.
The "position" on which you act is something different and not part of the force itself. This is connected to the notion of a force field, which tells you at each point $x \in \mathbb{R}^3$ the force $F(x)$ acting at that point. But at that point, it is a vector again.
A: 
I've noticed that textbooks usually assume that the tail of the force vector is where the force is being acted upon, but... if the tail of the 'vector' (force) is important and moving the force vector while keeping it parallel changes the outcome, then doesn't this mean that force is something more than a vector since it can't be moved around?

I think that you're asking about depictions like this,$\hspace{105px}$,where a force like $\color{red}{{\large{\textbf{N}}}\text{ormal force}}$ is drawn over multiple parts of space, right?
In other words, you're looking at diagrams in which forces are drawn out, and the graphical representation of a force appears to exist at multiple spatial points, right?
If so, then that's just a weirdness about how we have to draw stuff to get the information on paper.  For example, the $\color{red}{{\large{\textbf{N}}}\text{ormal force}}$ and $\color{blue}{{\large{\textbf{m}}}\text{ass}{\small{\times}}{\large{\textbf{g}}}\text{ravity}}$ in the image above are both actually just at that big black dot; they don't actually exist at any point in space besides that black dot.  Any apparent graphical representation to the contrary is just a drawing convention.
A: A force is a vector in the usual sense:
In physics a force can be considered as a quantity with direction and magnitude. But also forces satisfy the mathematical definition, i.e. they are elements of a vector space (meaning basically that they satisfy meaningful addition rules and can be scaled by multiplication with a number).
A common example for a force could be you throwing a ball. Before you let go of the ball, you apply a certain force in a certain direction, which can be written as a vector.
So much for what a force is. However your question seems to be about something a bit different. If you have an extended body, you correctly notice that the point where the force is applied will influence the motion of the body.
A very common model for such situation is the rigid body (basically a body that cannot be deformed). If you analyze the dynamics of such body under a force, you basically think of the body as consisting of many (infenitesimally) small masses, write down the usual equations of motion based on Newton and will find that quantities like torque and angular momentum will appear in the equations naturally. Both angular momentum and torque contain the information about the point of where the force is applied.
I suggest you to read up on rigid body dynamics and see if that helps to answer your questions.
