basic question of higher homotopy groups Homotopy groups of n-spheres are about embedding n-spheres into a manifold of dimension k. 
I want to understand what does operationally mean $\pi_n (S^k)$ when $n >k$. The definition of embedding i'm familiar with requires that the embedded manifold (in this case n-spheres) are always lower or equally dimensional.
In short, i don't have any intuition whatsoever what does it mean to embed a 2-sphere inside, say, a 1-sphere (a loop). What kind of mapping would that be?
 A: John has it right: the mistake is in the word "embedding".
Homotopy groups are defined as (pointed) homotopy classes of (pointed) maps from a sphere into a space (which does not have to be a manifold).
It turns out that there is no nontrivial way to map a $2$-sphere to a circle, but after that things get interesting...


*

*Consider the map $S^3 \rightarrow \mathbb{C}P^1$ where we consider $S^3 \subset \mathbb{C}^2 - \{0\}$. This is not nullhomotopic, and so it is a nontrivial way to map a $3$-sphere to $\mathbb{C}P^1$. (This is called the Hopf fibration, after identifying the target with $S^2$). One way to prove this is to show that the fibers over two different points are linked nontrivially.

*In general, the problem of calculating the homotopy groups $\pi_{n+k}S^{n}$ for $n$ large is equivalent to studying what is called "framed cobordism of $k$-manifolds". That is, we can say something about how complicated a map from a big sphere to a small sphere is by studying it's fiber which will (after a bit of wiggling) be some $k$-manifold. Saying that the map is nontrivial translates into saying that the manifold you get this way is nontrivial in some specific manner (called cobordism). 

*The above problem is hard and is, in some sense, the (unattainable) holy grail of algebraic topology: if we just cared about embeddings of spheres, our jobs would be over (or we'd become knot theorists...). Many nontrivial maps are known, and it is known that there a lots and lots.

