Two continuous functions are called homotopic if one of them can be continuously deformed into the other. Specifically, for continuous functions $f,g: X \to Y$, $f$ is homotopic to $g$, written $f \simeq g$, if there exists some continuous $H: X \times [0,1] \to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$.
A homotopy equivalence is then a map $f:X \rightarrow Y$ admitting a "homotopy inverse", i.e. a map $g:Y \rightarrow X$ such that $g \circ f \simeq \mbox{id}_X$ and $f\circ g \simeq \mbox{id}_Y$. Broadly speaking, then, homotopy theory is the study of topological spaces up to homotopy equivalence. As always, when one chooses to ignore certain aspects of the objects under study, other properties come to the fore. The first example of a homotopy-invariant property is that of the fundamental group.