What does the word "ideal" mean in this context? I'm confused about a terminology.
In Frank W. Warner's book Foundations of Differentiable Manifolds and Lie Groups, it says on page 12

Let $F_m$, a subset of $\bar{F_m}$ (the set of germs at $m$), be the set of germs which vanish at m. Then $F_m$ is an ideal in $\bar{F_m}$, and we let $F_m^k$ denote its kth power. $F_m^k$ is the ideal of $\bar{F_m}$ consisting of all finite linear combinations of k-fold products of elements of $F_m$.

Is an ideal this thing?
What does it mean to take the $k$th power of an ideal?

(this should really be a comment but, embarrassingly, I can't figure out how to make one.)
As others have mentioned, the term ideal comes from ring theory.  It's analogous to a normal subgroup - if you know group theory.  Do see the wikipedia page.  You can take products of two ideals $I$ and $J$ - it's the ideal $IJ$ generate by all products $xy$ for $x \in I$ and $y \in  J$.  The kth power of an ideal is simply the product with itself k times.  
You don't necessarily need a course in ring theory to read Warner, but it's helpful to have a friendly algebra book nearby for the things you haven't seen before.  My favourites are Abstract Algebra by Dummit and Foote and Algebra by Artin.  
