In this article , they write $$ df = f_x dx + f_y dy$$ and call it the 'differential of f' but what is a differental actually?
I know the directional directive is,
$$ D_{v} f = \nabla F \cdot v$$
And the gradient is $ \nabla F$ but what exactly is this mysterious differential quantity? Is it a one form? because I don't understand enough differential geometery to get that. But, I've seen this 'differential' idea used everywhere , especially in 'thermodynamics'. I have seen this post on mathestack exchange, however, I feel like I didn't understand anything after reading it. So, I'm looking for simpler explanations with more concrete examples.
It is indeed a 1-form. I'll offer a brief explanation here, but first let me suggest a couple of textbooks for a more leisurely treatment. First we have Gravitation by Misner, Thorne, and Wheeler, Part III, especially section 9.4. Don't be put off by this being a text on General Relativity; you also don't need to read the earlier sections. MTW (as it's often called) spill a lot of ink trying to give geometrical intuition.
Another good book: Applied Differential Geometry by William Burke.
OK, formally a 1-form is a smooth field of cotangent vectors, and a cotangent vector is a linear function on the space of tangent vectors. Let's unwrap that with a concrete example. Consider the function $f(x,y)=x^2+y^2$, the square of the distance from the origin. Pick a point in the plane, say at $P=(a,b)$. A tangent vector at $P$ is just a vector that we picture anchored at $P$. So think of an arrow going from $(a,b)$ to $(a+h,b+k)$. Let's say $v=(h,k)$. The coordinates of the tangent vector are just $(h,k)$, since the space of all tangent vectors at $P$ all have the same $(a,b)$.
Now we ask, how quickly is $f$ increasing in direction $v$? As you know, the answer is $D_v f = \nabla f\cdot v = f_x h+ f_y k$, which is $2ah+2bk$ here. That's a linear function of the tangent vector $(h,k)$. Such a linear function is called a cotangent vector.
We have a cotangent vector at every point in the plane, or in other words, a field of cotangent vectors. The cotangent vectors depend smoothly on the point $P$, since the coefficients $2a$ and $2b$ are smooth functions of its coordinates $(a,b)$. So that's the 1-form $df$.
MTW recommends picturing the 1-form by thinking about "infinitesimal" tangent vectors and how they cross level curves. Take an "infinitesimal" neighborhood of $P$, and blow it up by looking at it with a microscope set to "infinite" magnification. Look at all the level curves of $f$. They will look like a bunch of parallel lines, because of the whole infinitesimal thing. (This is just a fanciful way of talking about limits and linear approximations.) The signed number of parallel lines crossed by an infinitesimal vector $v$ tells you the directional derivative. That's a linear function of $v$. So a 1-form at a point can be pictured as a bunch of closely-spaced parallel lines. The more closely spaced, the bigger the value of the 1-form.
There's a lot more to say, both formally and intuitively, but I'll leave that to the books. (Also, their diagrams are invaluable.)