The homotopy groups are an infinite series of (covariant) functors $\pi_n$ indexed by non-negative integers from based topological spaces to groups for $n>0$ and sets for $n=0$ $\pi_n(X,x_0)$ as a set is the set of all homotopy classes of maps of pairs $(D^n,\partial D^n)\to (X,x_0)$ that is, maps of the disk into $X$ taking the boundary to the point $x_0$ Alternatively, these can be thought of as maps from the sphere $S^n$ into $X$ taking a basepoint on the sphere to $x_0$ These sets are given a group structure by declaring the product of 2 maps $f,g$ to simply attaching two disks $D_1,D_2$ with the right orientation along part of their boundaries to get a new disk $D_1\cup D_2$ and mapping $D_1$ by $f$ and $D_2$ by $g$ to get a map of $D_1\cup D_2$ This is continuous because we required that the boundary go to a fixed point, and well defined up to homotopy.

If $f:X\to Y$ satisfies $f(x_0)=y_0$ then we get a homomorphism of homotopy groups $f^*:\pi_n(X,x_0)\to\pi_n(Y,y_0)$ by simply composing with $f$ If $g$ is a map $D^n\to X$ then $f^*([g])=[f\circ g]$

More algebraically, we can define homotopy groups inductively by $\pi_n(X,x_0)\cong\pi_{n-1}(\Omega X,y_0)$ where $\Omega X$ is the loop space of $X$ and $y_0$ is the constant path sitting at $x_0$

If $n>1$ the groups we get are abelian.

Homotopy groups are invariant under homotopy equivalence, and higher homotopy groups ($n>1$ are not changed by the taking of covering spaces.

Some examples are:

$\pi_n(S^n)=\mathbb{Z}$

$\pi_m(S^n)=0$ if $m<n$

$\pi_n(S^1)=0$ if $n>1$

$\pi_n(M)=0$ for $n>1$ where $M$ is any surface of nonpositive Euler characteristic (not a sphere or projective plane).