Is there a specific part of the definition you don't understand? From Wikipedia:
Given two spaces $X$ and $Y$, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps $f : X → Y$ and $g : Y → X$ such that $g ∘ f$ is homotopic to the identity map id $X$ and $f ∘ g$ is homotopic to id $Y$.
The article also gives a definition of homotopy between functions
Formally, a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space X with the unit interval $[0,1]$ to $Y$ such that, if $x ∈ X$ then $H(x,0) = f(x)$ and $H(x,1) = g(x)$.