How to view Stone-Cech compactification of the real line?

Question

I am going through Arveson's A Short Course on Spectral Theory and have come across an exercise constructing $\beta\mathbb{R}$ using the Gelfand map. I was wondering if there is an explicit characterisation of maximal ideals in $C_b(\mathbb{R})$ the space of bounded complex values continuous functions on $\mathbb{R}$? Clearly evaluation at any point in $\mathbb{R}$ is such a maximal ideal and corresponds to $\mathbb{R}\subset\beta\mathbb{R}$ but what about the others,the so called corona of $\mathbb{R}$? Every text I look up uses $\beta\mathbb{R}$ to characterise these but I don't think this is explicit?