What are the advantages / disadvantages of using splines, smoothed splines, and gaussian process emulators?

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I am interested in learning (and implementing) an alternative to polynomial interpolation.

However, I am having trouble finding a good description of how these methods work, how they relate, and how they compare.

I would appreciate your input on the pros/cons/conditions under which these methods or alternatives would be useful, but some good references to texts, slides, or podcasts would be sufficient.

note: I asked this question on stats.SE a few weeks ago and received the recommendation to ask here since I have not received an answer.

Thanks!