I thought a vector field was a function $\vec{A}:\mathbb{R}^n\to\mathbb{R}^n$, which takes a vector to a vector. This seemed intuitive to me, but in a mathematical physics course I found the definition
$$X=\sum\limits\_{i=1}^n X^i \frac{\partial}{\partial x^i},$$ 
where $X$ seems to take functions from $C^\infty$ as argument and $X^i:\mathbb{R}^n\to\mathbb{R}$. I don't understand this new way of talking about vector fields, especially when our professor said we should see the $\frac{\partial}{\partial x^i}$ as basis vectors of the tangent space. 

I also can't really translate any vector field which I know into this new definition. So it would be great if someone could maybe make an example of an $\vec{E}$-field or something similar and "translate" it into this new way of expressing vector fields. I'm also struggling to visualize the new definition. It seems that $X(f):\mathbb{R}^n\to \mathbb{R}$ (for $f\in C^\infty$), to which I really can't assign any physical meaning.

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**TL;DR:** Can someone explain to me why someone would want to express a vector field in this way? Ideally I'd like to see an example of the same vector field expressed in the two forms and an explanation how one would interpret the visualization of the new vector field.