2
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I have this code:

AllData = Import["https://pastebin.com/raw/TARH18Hm", "Table"];
Do[Spec[2*i] = AllData[[All, (2*i + 1) ;; (2*i + 2)]], {i, 0, 9}];
peakpositions = FindPeaks[Spec[18][[All, 2]], 3, 1, 400];
peakswavelength = Spec[18][[#1, 1]] & @@@ peakpositions;
peaks = Transpose[{peakswavelength, peakpositions[[All, 2]]}];
ListPlot[{Spec[18], Callout[#, Rotate[#[[1]], 0], Above] & /@ peaks}, 
PlotRange -> All, Joined -> {True, False}]

Which generates this:

enter image description here

I labeled five major peaks but I want to replace the numeric labels with five markers like a filled circle, a filled square and so on. How can I do that?

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The second argument to Callout can be any expression. In your case you are just using the wavelength (#[[1]]) as the label. To use different labels, you can do something like this:

labels = {"\[FilledSquare]",  Style["\[FilledUpTriangle]", Blue], 
          "\[FilledCircle]", "\[FilledDiamond]",
          Import @ "ExampleData/spikey.tiff"};
labeledPeaks = MapThread[Callout[#, #2, Above]&,
    {peaks, labels}
];
ListPlot[{Spec @ 18, labeledPeaks},
    PlotRange -> All,
    Joined -> {True, False}
]

enter image description here

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