Independence
Definition
Random variables $X$ and $Y$ are independent (written $X \perp Y$) if
$$ p(x, y) = p(x)\, p(y) \quad \text{for all } x, y. $$Equivalently, $X \perp Y$ iff $p(x \mid y) = p(x)$ for all $x, y$.
Conditional Independence
$X$ and $Y$ are conditionally independent given $Z$ (written $X \perp Y \mid Z$) if
$$ p(x, y \mid z) = p(x \mid z)\, p(y \mid z) \quad \text{for all } x, y, z. $$Independence does not imply conditional independence, and vice versa.