Is saying each pt of a topo. manifold has nbhd homeomorphic to R^n the same thing as saying there is a local coordinate system at each point? Is saying each point of a topological manifold has a neighborhood homeomorphic to $\Bbb{R}^{n}$ the same thing as saying there is a local coordinate system at each point?   

I'm not really sure what I mean by coordinate system. It's just a term I have heard used. If someone could define that at the same time that would be awesome.
Yes. This is true by definition:

A topological space $M$ is said to be locally Euclidean of dimension $n$ if each point $x \in M$ has a neighbourhood $U$ homeomorphic to an open subset $V$ of $\mathbb{R}^n$. A homeomorphism of this sort is by definition a coordinate chart.
generally when imposing coordinate systems there is the additional condition that they are compatible. Thus if two neighbourhood's intersect then the intersection the associated two homeomorphisms yield a map between two subsets of $R^n.$ This map must also be a homeomorphism for a coordinate system.