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I have the following set of time-series data that I would like to check for anomalies in:

enter image description here

What I would like to do is to calculate the curve from one and see if that curve is matched, within some range, against another curve. So visually what I want to achieve is the following: enter image description here Where we've taken the curve from the 2nd peak, and applied it to the data of the 1st peak. The curves diverge at one point, denoted by the red highlight. (in a real world scenario, I'd rather be applying the previous day's curve to today's).

I've looked into ARIMA but this is a foreign concept to me, so I'm not really sure where to start. What sort of models should I be looking into for this sort of analysis?

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A common approach is indeed to first model the data using some time series model, make a prediction, and then evaluate how far the current measurement deviates from the prediction (or try a cumulative error over several time points). It is possible to make a distribution of the errors from the fitted data.

This approach stands and falls with the series being predictable. Try Holt-Winters instead of ARIMA. This is supported in R. For a plug and go python implementation see: https://gist.github.com/andrequeiroz/5888967

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