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I am struggling to align the content of this table. Please, could you kindly provide some help?

 begintable[]
 centering
 captiontextcolorredLaminates devised for in-plane (tension and compression) and mode-I interlaminar fracture toughness experiments. Non-crimp fabric blankets are indicated with parenthesis. The $$$ sign means only one veil in the symmetry plane. Explicar y poner el capitulo correspondiente. Poner bonita
 labeltable:notation
 begintabularxlinewidth@ l 
 >hsize=0.65hsizeX % <---
 >hsize=0.35hsizeX % <---
 @
 toprule
 Laminate & Stacking sequence & Purpose \ midrule
 LTHINtextsubscriptILSS & multirow3*$[(45degree/0degree)/(-45degree/90degree)]textsubscript4s$ & Interlaminar shear strenght testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LTHINtextsubscriptIP & & Tensile and compressive testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LTHINtextsubscriptLVI & & Impact and quasi-static indentation testing \ midrule
 LTHINtextsubscript$G$textsubscriptIC & $[(45degree/90degree)/(-45degree/0degree)]textsubscript4s$ & Mode-I interlaminar fracture toughness testing \ midrule
 LSTANDARDtextsubscriptILSS & multirow2*$[(45degree/0degree)/(-45degree/90degree)]textsubscript2s$ & Interlaminar shear strenght testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LSTANDARDtextsubscriptLVI & & Impact and Quasi-static indentation testing \ midrule
 LV1textsubscriptIP and LV2textsubscriptIP & multirow2*$[$(45$degree$/V$_i$/0$degree$)/V$_i$/($-$45$degree$/V$_i$/90$degree$)/V$_i$$]$textsubscript4$$$ ($i$ = 1,2) & Tensile and compressive testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LV1textsubscriptLVI and LV2textsubscriptLVI & & Impact and Quasi-static indentation testing \ midrule
 LV1textsubscript$G$textsubscriptIC and LV2textsubscript$G$textsubscriptIC & $[$(45$degree$/90$degree$)/($-$45$degree$/0$degree$)/V$_i$$]$textsubscript6$$$ ($i$ = 1,2) & Mode-I interlaminar fracture toughness testing \ bottomrule 
 endtabularx
endtable

like this:

documentclass[12pt,a4paper]article
usepackagebooktabs, multirow, tabularx
newcolumntypeC>centeringarraybackslashX
newcolumntypeL>raggedrightarraybackslashX
usepackagesiunitx
usepackage[skip=1ex, 
 textfont=color=red]caption

%---------------- show page layout. don't use in a real document!
usepackageshowframe
renewcommandShowFrameLinethickness0.15pt
renewcommand*ShowFrameColorcolorred
%---------------------------------------------------------------%

begindocument
 begintable
renewcommandtabularxcolumn[1]m#1% valid local for tis table
centering
captionLaminates devised for in-plane (tension and compression) and mode-I interlaminar fracture toughness experiments. Non-crimp fabric blankets are indicated with parenthesis. The $$$ sign means only one veil in the symmetry plane. Explicar y poner el capitulo correspondiente. Poner bonita
labeltable:notation
begintabularxlinewidth@ l C L @
 toprule
Laminate & textStacking sequence & Purpose \
 midrule
LTHINtextsubscriptILSS 
 & multirow6=centering
 [(SI45degree/%
 SI0degree)/%
 (SI-45degree/%
 SI90degree)]textsubscript4s
 & Interlaminar shear strenght testing \ 
 cmidrule1-1 cmidrule3-3
LTHINtextsubscriptIP 
 & & Tensile and compressive testing \ 
 cmidrule1-1 cmidrule3-3
LTHINtextsubscriptLVI 
 & & Impact and quasi-static indentation testing \ 
 midrule
LTHINtextsubscript$G$textsubscriptIC 
 & 
 [(SI45degree/%
 SI90degree/%
 (SI-45degree/
 SI0degree)]textsubscript4s
 & Mode-I interlaminar fracture toughness testing \ 
 midrule
LV1textsubscriptIP and LV2textsubscriptIP 
 & multirow3.4=centering
 [(SI45degree/V$_i$/SI0degree)/ 
 V$_i$/(SI-45degree/%
 V$_i$/SI90degree)/%
 V$_i$]textsubscript4$ ($i = 1,2$) 
 & Tensile and compressive testing \ 
 cmidrule1-1 cmidrule3-3
LV1textsubscriptLVI and LV2textsubscriptLVI
 & & Impact and Quasi-static indentation testing \ 
endtabularx
 endtable
enddocument

table is not complete, however it show how to solve all your requirements (as i understood them). for values with units i use the siunitx package, and for caption the caption package. i also change width of columns. i suggest to have equal width of the second and third column. for long sequence, which not fit in cell i enable that it can be broken into three lines

You will most likely get mad at me but I really think you will greatly benefit from trying to understand what math vs. text mode are good for, how simple it is to get a subscript in math mode (vs. text mode) and so on. I leave all these things untouched and focus on the multirows, which I replaced by multirowcells that come with the makecell package.

documentclassarticle
usepackagetabularx
usepackagebooktabs
usepackagemultirow
usepackagemakecell
%usepackagesiunitx
defdegreeensuremath^circ
usepackagexcolor
begindocument
begintable[]
 centering
 captiontextcolorredLaminates devised for in-plane (tension and compression) and mode-I interlaminar fracture toughness experiments. Non-crimp fabric blankets are indicated with parenthesis. The $$$ sign means only one veil in the symmetry plane. Explicar y poner el capitulo correspondiente. Poner bonita
 labeltable:notation
 begintabularxlinewidth@ l 
 >hsize=0.65hsizeX % <---
 >hsize=0.35hsizeX % <---
 @
 toprule
 Laminate & Stacking sequence & Purpose \ midrule
 LTHINtextsubscriptILSS & 
 multirowcell3[-5ex][l]$[(45degree/0degree)/(-45degree/90degree)]textsubscript4s$ 
 & Interlaminar shear strenght testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LTHINtextsubscriptIP & & Tensile and compressive testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LTHINtextsubscriptLVI & & Impact and quasi-static indentation testing \ midrule
 LTHINtextsubscript$G$textsubscriptIC & 
 multirowcell2[-1ex][l]$[(45degree/90degree)/(-45degree/0degree)]textsubscript4s$
 & Mode-I interlaminar fracture toughness testing \ midrule
 LSTANDARDtextsubscriptILSS & 
 multirowcell2[-3ex][l]$[(45degree/0degree)/(-45degree/90degree)]textsubscript2s$ & Interlaminar shear strenght testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LSTANDARDtextsubscriptLVI & & Impact and Quasi-static indentation testing \ midrule
 LV1textsubscriptIP and LV2textsubscriptIP & 
 multirowcell3[-3ex][l]$[$(45$degree$/V$_i$/0$degree$)/V$_i$/($-$45$degree$/V$_i$/90$degree$)/
 \ quad V$_i]$textsubscript4$$$ ($i$ = 1,2) & Tensile and compressive testing \ cmidrule(r)1-1 cmidrule(r)3-3
 LV1textsubscriptLVI and LV2textsubscriptLVI & & Impact and Quasi-static indentation testing \ midrule
 LV1textsubscript$G$textsubscriptIC and LV2textsubscript$G$textsubscriptIC & 
 multirowcell2[-1ex][l]$[$(45$degree$/90$degree$)/($-$45$degree$/0$degree$)/V$_i$$]$textsubscript6$$$
 \ ($i$ = 1,2) & Mode-I interlaminar fracture toughness testing \ bottomrule 
 endtabularx
endtable
enddocument

And please consider providing a complete compilable document in the future. Here I just quickly made up something crappy for degree but I would like to argue that you should not use a macro of this name unless it is from the siunitx package.