Please correct me if I'm wrong, which is highly likely.
Say I have $f:U\to \mathbb{R^n}$ ($f$ is a differenciable function and $U\subset \mathbb{R^m}$). $f$ is an immersion if $rank(f,x)=m$. From what I gather then an immersion is some function that takes a bunch of vectors from $U$ to $\mathbb{R^m}$ and still preserves $U$'s characteristics, even though it is in another dimension. For example if I take the whole line of $\mathbb{R}$ and stick it into $\mathbb{R^2}$ under the form of a unitary circunference it is an immersion as it still has the "characteristics of a line" or that it's still a line. However, this is the case where I go from lower dimensions to higher one. If my line of thought is correct then there are no immersions that go from a higher dimension to a lower one, hence in order for $f$ to be an immersion $m\leq n$.Then I could also think of "immersing" an entire circle onto $\mathbb{R^3}$ so that it would still be a plane.
Now for a smooth embedding. It is defined as in injective immersion whose inverse is continuous. The example I have is a simple one. Suppose I have a rope of lenght $4\pi$ and I decide to twist it to form a circunference of radius one, then the rope overlaps itself, or one could also say that it self-intersects, hence when we formalize this, we can say that the function $f:]0;4\pi[\to \mathbb{R^2},x\to (cos(x),sin(x))$ is not injective, hence it is not a smooth embedding. But now we cut the rope to only $2\pi$ and do the same process. Now the rope doesn't self intersect and $f$ is injective with a continuous inverse, hence it is a smooth embedding (Why is there a need for a continuous inverse?). So a smooth embedding is a mapping that cannot self intersect (?).This is a visual and easy example to grasp, but it is again the case where we go from lower to higher dimensions, what about going from higher to lower ones? Do they exist?
I came here to ask you for some visual and fundamental examples for that is what I cannot find in books (or perhaps I haven´t found the right ones) or anywhere on the internet, please leave the formalisms aside. I believe that this can be a very geometrical and visual discipline, so please bear with me.