Bound variables refer to variables that serve as a placeholder to point to a binder. In calculus, "∫ t dx" is a binder and any free appearance of the variable "x" in "t" is bound.
Take a term a in the context $(\Gamma,x{:}T)$ and apply some operation $f$ to it to create a term $f(a)$ (possibly of a different type) in the context $\Gamma$; then any appearances of the variable $x$ in the term $a$ have become bound variables in the term $f(a)$. (from nLab)
Related tags:
binders
free-variables
References:
nLab
Wolfram MathWorld
ProofWiki
Encyclopedia of Mathematics
