Intuition of a Submanifold Could someone explain the intuition behind a submanifold. When, for example, is it appropriate to work with immersed submanifolds vs embedded submanifolds? Why is it important for a submanifold to be say smooth and in the class $C^\infty$. For the project I'm completing, I'm working with manifolds, but like I said previously: 
https://math.stackexchange.com/questions/560940/how-to-make-a-ghost-manifold 
I need the manifolds to be able to pass through each other, and it is my understanding that by default submanifolds have this property. Why so? What about if one is a submanifold and the other is a manifold? Do they still have this property?
 But after that, I do not know which type of manifold to use, or what differential class to put them in.   
If any user with vast knowledge of topology would allow myself to email them a pdf that would give them a better understanding of what I am after, it would be greatly appreciated.
 A: Answering your question properly is impossible without knowing your background: I knew some 10th graders who already took advanced graduate classes. Here is the background you need in order to get true understanding of differentiable manifolds:
Calculus (through vector calculus); some linear algebra (vector spaces, subspaces, linear maps, determinants); rigorous real analysis; some point-set (general) topology (you need to get comfortable with the concept of a topological space, compactness and quotient topology). 
If you have this background, keep reading below. Otherwise, I suggest you pick up something like Rudin's "Principles of Mathematical Analysis". If this is too advanced, try Courant and Robbins "What is Mathematics?" This is the best math book I can recommend to a HS student. 
Now, there are several excellent textbooks covering differential topology on the undergraduate level:


*

*Guillemin and Pollack "Differential Topology"

*Singer and Thorpe "Lecture Notes on Elementary Topology and Geometry". 
If you have the right background, you will learn much more from reading these books than from anything I (or anybody else) can write here, at the MSE. 
Now, assume that you at least started reading either 1 or 2, but you still have questions about importance of differentiability, submanifolds, and so on, that you raised in your question. Here is my take:
There are three principal kinds of manifolds topologists study: Topological manifolds, piecewise-linear (PL) manifolds and differentiable manifolds. For the first two types of manifolds, you do not assume (and you do not need) any differentiability whatsoever. Then why study smooth manifolds? Several reasons. One is that you can use tools not available to you otherwise, loosely speaking, you can use (vector) calculus/real analysis of several variables. Secondly, smooth manifolds tend to appear in many areas of applications of topology, e.g., in differential geometry, mathematical physics, differential equations, algebraic geometry, and so on.  
Why do you need $C^\infty$ if you study smooth manifolds? Most of the time, you do not really need infinite differentiability; what you need is "differentiable as many times as needed". Without this, one gets into some delicate analytical issues which, as a topologist, you probably would prefer to avoid. For instance, to do differential geometry you typically need $C^2$-smoothness. On the other hand, in order to use Sard's theorem for maps $f: M^m\to N^n$, you need $f$ to be $C^k$ with $k\ge \max\{n-m+1, 1\}$. In order to avoid keeping track of this, one (in many cases) simply assumes that everything (manifolds and maps) is infinitely differentiable. 
Why study submanifolds? For instance, because they appear naturally as "generic" solutions of equations, which is what Sard's theorem is about (think of the solution set of the equation $f(x)=a$, where $f: M\to {\mathbb R}^n$, $a\in {\mathbb R}^n$). I hope that you do not have to be convinced that equations are important (much of mathematics is about equations, in one form or another). Another way to think of submanifolds is as generalizations of graphs of (smooth) functions of several variables. Now, to your question about submanifolds intersecting each other. You should ask yourself: Can graphs of two functions intersect each other? (Hint: Think of linear functions. Can their graphs intersect?) Can graphs of functions pass through each other? Think of the functions $f(x)=x^2$, and $g(x)=a$, where you treat $a$ as a parameter which can vary. Do the graphs of these functions intersect when $a<0$? What about $a>0$? 
Now, about "immersed submanifolds": This is just a slang which is an abbreviation for "An immersion $f: M^m\to N^n$." If you are just starting to study topology, it is best to avoid this slang. Then there are only immersions (maps) and submanifolds (subsets). 
Lastly, just forget about asking somebody at MSE (or mathoverflow) to read your text: Type it, print it, bind it if you like and put it on the shelf. I did something like this when I was 17. I find it now amusing but not more than that. 
