Please take a quick look at the documentation for Convolve:

reference.wolfram.com/mathematica/ref/Convolve.html

Convolve requires 4 arguments.  Additionally the examples show that it is meant to take symbolic functions, not numerical functions.

I don't want to complicate the math of this at all. But I should warn against using too many tools like interpolations if they are not needed. If you can resample the data set to keep it discrete and do a discrete convolution, I would recommend it instead of continuing down the path below.

Here's the definition of a convolution. I will use this with a couple of example InterpolationFunction expressions I've constructed.



"f" and "g" are only defined from x = 0 to 10, so this will change the range of the integration. In short the integral is (I'm mostly sure):
"y" of course must be between 0 and 10 for this to make sense.
Because f and g are not symbolic, they are numeric, we will have to use NIntegrate:
See this article for why the function was defined with ?NumericQ.

This tutorial shows more about how to do convolutions and correlations.
 
Yes the documentation shows Kernel, but that can just be another list. Both ListCorrelate and ListConvolve have an argument that changes the shift on the operation. Consider this code for example:
Which gives:

It might be helpful to know and think about what tools your advisor has used in the past for this kind of problem. The tools they used will probably inform how they describe the process that they're looking for. I remember this being an issue for me with my internships.  My advisors often communicated what they wanted using the process and format of old programs that they used. More often than not, the hardest part of the process was piecing together in formal terms, usually mathematical, what was needed.

I'm not sure exactly how accurately I can tell what needs to be done here. I'm pretty certain that you will have to go back and ask for more precise details about what should get done. More important than understanding the steps is understanding the goal. You will probably need to become decently familiar with programming in Mathematica to program a way to achieve that goal.

First , it looks like your data comprises of a pairs of wavelengths and insentities (amplitudes?). I am not sure how these pairs relate to the original signal you want to study. Before doing a Fourier Transform or anything, you'll want to convert whatever form your data is in into a time series of the actual signal. You mentioned that the sampling isn't even. If that is the case, you will probably want to do some kind of interpolation and then resample at an even sample rate. You can use Mathematica's Interpolation function or write a custom interpolation scheme depending on your needs. Once you have both signals at the same sample rate, use ListConvolve to convolve the lists and compute an auto correlation. You can vary the offset used in the function to find the optimial offset.