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After reading some answers on the site, I'm trying to use dcolumn pacakge in order to add horizontal alignment to my tables.
Unfortunately without success so far.
I'm writing as a part of a group so I shouldn't added packages that may affect others, that's why I'm trying to figure it w/o adding the siunitx package.

documentclass{article}

usepackage[utf8]{inputenc}

usepackage{booktabs}

usepackage{multirow}

usepackage{ctable}

usepackage{tabularx}

usepackage{graphicx}

usepackage{dcolumn} 

newcolumntype{d}[1]{D{.}{.}{#1} }





begin{document}



begin{table}

    centering

    begin{tabular}{@{}lc*{4}{d{1.5}}@{}}

        toprule

        {} & {} & multicolumn{2}{c}{Group A} & multicolumn{2}{c}{Group B} \

        cmidrule(lr){3-4}  cmidrule(l){5-6}

        Types & Character & Results A & Results B     & Results A & Results B     \

        midrule

        multirow{3}{*}{Main}

        {}    & The Good & .111      & .789_{b}^{b}  & .520      & .555_{b}^{a}  \

        {}    & The Bad  & .111      & .636_{b}^{al} & .520      & .730_{b}^{al} \

        {}    & The Ugly & .111      & .525_{b}      & .520      & .696_{b}      \

        bottomrule

    end{tabular}

    caption{Caption}

    label{tab:my_label}

end{table}

The result is:

Please help me solve this puzzle, also I need to add bold to some values in the table, and since switching to dcolumn I've removed the $n$ I've had and therefore mathbf seems not to work any more. What would be the solution in that case?

You are probably looking for multicolumn:

documentclass{article}

usepackage[utf8]{inputenc}

usepackage{booktabs}

usepackage{multirow}

usepackage{ctable}

usepackage{tabularx}

usepackage{graphicx}

usepackage{dcolumn}

newcolumntype{d}[1]{D{.}{.}{#1} }





begin{document}



begin{table}

  centering

  begin{tabular}{@{}lc*{4}{d{1.5}}@{}}

    toprule

    {} & {} & multicolumn{2}{c}{Group A} & multicolumn{2}{c}{Group B} \

    cmidrule(lr){3-4}  cmidrule(l){5-6}

    Types & Character & multicolumn{1}{c}{Results A} & multicolumn{1}{c}{Results B} & multicolumn{1}{c}{Results A} & multicolumn{1}{c}{Results B}  \

    midrule

    multirow{3}{*}{Main}

    {}    & The Good & .111      & .789_{b}^{b}  & .520      & .555_{b}^{a}  \

    {}    & The Bad  & .111      & .636_{b}^{al} & .520      & .730_{b}^{al} \

    {}    & The Ugly & .111      & .525_{b}      & .520      & .696_{b}      \

    bottomrule

  end{tabular}

  caption{Caption}

  label{tab:my_label}

end{table}



end{document}

However, as mentioned in the comments, I think you should reconsider the case against siunitx. More important than being a "basic package" is the fact that it is very well crafted and zealously maintained, besides being widely used. (This is not a judgement on dcolumn of which I'm not an user). You should, of course, decide that along with your work group.

The same table done with siunitx would be something like:

documentclass{article}



usepackage{booktabs}

usepackage{multirow}

usepackage{siunitx}



begin{document}



begin{table}

  centering

  sisetup{table-format = 0.3}

  begin{tabular}{@{}

    lc

    S

    S[table-space-text-post = {$_{b}^{al}$}] % make it the largest post-number element

    S

    S[table-space-text-post = {$_{b}^{al}$}]

    @{}}

    toprule

          &           & multicolumn{2}{c}{Group A}     & multicolumn{2}{c}{Group B}    \

    cmidrule(lr){3-4}  cmidrule(l){5-6}

    Types & Character & {Results A} & {Results B}       & {Results A} & {Results B}      \

    midrule

    multirow{3}{*}{Main}

          & The Good  & .111        & .789{$_{b}^{b}$}  & .520        & .555{$_{b}^{a}$}  \

          & The Bad   & .111        & .636{$_{b}^{al}$} & .520        & .730{$_{b}^{al}$} \

          & The Ugly  & .111        & .525{$_{b}$}      & .520        & .696{$_{b}$}      \

    bottomrule

  end{tabular}

  caption{Caption}

  label{tab:my_label}

end{table}



end{document}

It does automatically take care of the leading zero, emphasized by @Denis in the comments.

You can (should, in my opinion) use siunitx; here I show also how to have real note markers upright that leave the exact space for them.

If you want to set some of the entries bold, there is a trick adapted from https://tex.stackexchange.com/a/352028/4427

documentclass{article}

usepackage[utf8]{inputenc}

usepackage{booktabs}

usepackage{siunitx}



NewDocumentCommand{bs}{}{fontseries{b}selectfont}



makeatletter

NewDocumentCommand{tss}{mm}{%

  {m@thensuremath{%

    ^{mbox{fontsizesf@sizez@selectfont #1}}%

    _{mbox{fontsizesf@sizez@selectfont #2}}%

  }}%

}

makeatother



begin{document}



begin{table}[htp]

centering



sisetup{detect-weight,mode=text}

DeclareDocumentCommand{bfseries}{}{bs}



begin{tabular}{

  @{}

  l

  c

  S[table-format=1.3]

  S[table-format=1.3,table-space-text-post=tss{al}{b}]

  S[table-format=1.3]

  S[table-format=1.3,table-space-text-post=tss{al}{b}]

  @{}

}

toprule

 & & multicolumn{2}{c}{Group A} & multicolumn{2}{c}{Group B} \

cmidrule(lr){3-4}  cmidrule(l){5-6}

Types & Character & {Results A} & {Results B} & {Results A} & {Results B} \

midrule

Main & The Good & bs .111 & .789tss{b}{b}  & .520 & .555tss{a}{b}  \

     & The Bad  &     .111 & .636tss{al}{b} & .520 & .730tss{al}{b} \

     & The Ugly &     .111 & .525tss{}{b}   & .520 & .696tss{}{b}   \

bottomrule

end{tabular}



caption{Caption}

label{tab:my_label}



end{table}



end{document}

I wouldn't use multirow: blank space below “Main” means repetition. Shifting it down makes quite unclear whether “Main” refers to all rows or just the middle one.