Since $l$ and $m$ represent two different lines, where $m$ is within the plane $\alpha$,
if "$l \perp m$", we can have $m \subset \alpha$, but we cannot deduce $l \perp \alpha$ from this;
if "$l \perp \alpha$", then $l$ is perpendicular to every line within the plane $\alpha$, so $l \perp m$,
Therefore, "$l \perp \alpha$" implies "$l \perp m"
Thus, "$l \perp m$" is a necessary but not sufficient condition for "$l \perp \alpha$",
Hence, the correct choice is $\boxed{\text{B}}$;