Regularly homotopic vs just homotopic - telling apart with normal bundles... I know that given a regularly homotopic class immersions of a manifold into another manifold $f,g:N^n \to M^m$, then the resulting normal bundles $\nu_f,\nu_g : N \to BO(m-n)$ are the same (i.e. $\nu_f \simeq \nu_g$) - this gives rise to the normal bundle homomorphism $\nu : I_n(M) \to \pi_n(BO(m-n))$ where $I_n(M)$ is the set of regular homotopy classes of immersions $S^n \to M$.
When I think about this I usually make the mistake of thinking that only the homotopy class matters, and not the normal homotopy class - for example this would give a map $\pi_n(M) \to \pi_n(BO(m-n))$ that the above would factor through.  I imagine that this is totally false - but I couldn't think of any nice examples.
What are some examples of immersed spheres that are homotopic but not regularly homotopic?  What are some examples where we see they are not regularly homotopic because they have different normal bundles?
 A: Milnor proves that in a manifold of dimension $2m$ such that the tangent bundle restricted to the m-skeleton is trivializable, an embedded m-sphere has trivial normal bundle, iff, the class it represents in homology pairs with itself to 0.
For example, since the torus is parallelizable we can take an embedded circle and see that it must have trivial normal bundle since any elements of $H^1 (T)$ squares to 0.
In my opinion, the most impressive use of these definitions and results involving immersions come in the form of surgery theory. In surgery theory, we look to alter manifolds by very drastic means (cutting and gluing) in order to obtain a manifold that suits us better.
The way surgery goes is that you take a copy of $S^p \times D^{q+1}$ inside your $p+q+1$ dimensional manifold $M$, cut it out, and then glue in $S^q \times D^{p+1}$ via the canonical identification of $S^p \times S^q \cong S^q \times S^p$. The first important thing to notice is that if $p<q$, we have essentially killed the homotopy class in $M$ associated to the $S^p$ and only affected higher homotopy groups. So one should think how we could use this to our advantage.
We would like to be able to kill whatever homotopy groups we want, but the first obstruction is that the homotopy class is represented by an embedding (which in the dimensions we care most about is possible) and then this embedding has to have trivial normal bundle (in order to find the disk needed in the definition of surgery). In the odd dimensional case, this ends up being easy (again in the situations that we want to do surgery on; these are things like killing the first n homotopy groups of a 2n+1 dimensional parallelizable manifold or killing the first n relative homotopy groups of a map covering the normal bundles).
However, in the even dimensional case it is not always possible. If it were, we could easily surgically convert any even dimensional manifold into a (homology) sphere, but there is at least one obstruction to this: the signature of the manifold. It is easy to see that surgery does not alter the signature of the manifold, so if your manifold has nontrivial signature than it can't be surgically converted to a sphere. Perhaps this is what caused Milnor to look for a pure homological description of when a embedding should have trivial normal bundle.
If embeddings and surgery sound interesting to you, I strongly recommend learning some surgery theory. There are many good accounts: Milnor's "A Procedure for Killing Homotopy Groups of Differentiable Manifolds" is a good introduction if you are pretty sure you'll be interested. If you just want a taste, Browder has the very short "Homotopy Type of Differentiable Manifolds" which gives an overview of the uses of surgery and sketches part of a homotopy classification of manifolds. Another classic (which you should definitely read at some point) is Kervaire and Milnor's "Groups of Homotopy Spheres I". It is a beautiful blend of homotopy and geometry. This basically contains most of the elements of simply connected surgery theory.
If all of those look good to you, then you should check out a textbook. I currently use "Geometric and Algebraic Surgery" by Ranicki, but Wall has a good textbook as well. If all of this is a little to intense, it might be a good idea to just look at the classification of immersions $S^1 \rightarrow \mathbb{R}^2$.
I'll add to specifically address your question: Milnor and Kervaire in their paper specifically study the difference between homotopy classes of embeddings/immersions and regular homotopy classes. They do this in order to show that you can always find a class to do surgery on to get you one step closer to being a sphere (again this is in the context of surgery on parallelizable manifolds).
