What does it exactly mean for a subspace to be dense? My understanding of rationals being dense in real numbers:
I know when we say the rationals are dense in real is because between any two rationals we can find a irrational number. In other words we can approximate irrational numbers using rationals. I think a more precise definition would be is that any open ball around a irrational number will contain a rational number. 
If what I said is correct, I am trying to think about what it means for $C[a,b]$ (which are the continuous complex valued functions on [a.b]) to be dense subspace of $L^2[a,b]$. From what I said above, I want to say that all functions in $L^2[a,b]$ can be approximated by functions from $C[a,b]$. Is the intuition correct here, what would the precise definition in this case?

In general topological spaces, a dense set is one whose intersection with any nonempty open set is nonempty. For metric spaces, since we have a topological base of open balls, this is equivalent to every point in space space being arbitrarily close, with regards to the metric, to point in the dense set.
Note that $L^2$ is a metric space, where $d(f,g) = ||f-g||_2$, with $||\cdot||_2$ being the $L^2$ norm.
