Ttdfjt

I would like the formulae in the third column to be vertically aligned to the middle, i.e. the formulae should be vertically at the middle of the two lines of the descriptions.

documentclassarticle
usepackageamsmath
usepackagetabularx,booktabs
usepackagecaption
begindocument
begintable[htb]
 centering
 captionI would like the formulae here to be vertically at the middle of the two lines of text
 begintabularxlinewidthlX>$c<$
 toprule
 Name & Description & multicolumn1cFormula\
 midrule
 Ducks & Something which goes on two lines, something which goes on two lines &
 A=dfracBC\
 Lions & Something which goes on two lines, something which goes on two lines &
 D=dfracEF\
 bottomrule
 endtabularx
endtable
begintable[htb]
 centering
 captionHere the formulae are correctly aligned with the description, but I would like the name at the top like in the previous table
 begintabularlm24em>$c<$
 toprule
 Name & Description & multicolumn1cFormula\
 midrule
 Ducks & Something which goes on two lines, something which goes on two lines &
 A=dfracBC\
 Lions & Something which goes on two lines, something which goes on two lines &
 D=dfracEF\
 bottomrule
 endtabular
endtable
enddocument

Here a solution based on the redefinition of the X column type, and two possible hacks for the first column (they may have to be adapted to the real contents):

documentclassarticle
usepackageamsmath
usepackagetabularx, booktabs, makecell
renewcommandtabularxcolumn[1]m#1

begindocument

begintable[htb]
 centering
 begintabularxlinewidthlX>$c<$
 toprule
 Name & Description & multicolumn1cFormula\
 midrule
 makecell[l]Ducks\mbox & Something which goes on two lines, something which goes on two lines &
 A=dfracBC\
addlinespace
 makecell[l]Lions\mbox & Something which goes on two lines, something which goes on two lines &
 D=dfracEF\
addlinespace
 raisebox1.4exLions & Something which goes on two lines, something which goes on two lines &
 D=dfracEF\
 bottomrule
 endtabularx
endtable

enddocument 

This is rather hard: the problem is that you want to align the middle box in two ways: on the left along the baseline, on the right along the center. This type of table normally requires either nesting of tables (which is difficult with tabularx) or measuring some of the content to use llap tricks.

We are missing here tabular code based on xcoffins which has more than one handle ...

documentclassarticle
usepackageamsmath
usepackagetabularx,booktabs
usepackagecaption,xcoffins
newlengthcolA

begindocument
begintable[htb]
 centering
 renewcommandtabularxcolumn[1]m#1
 settowidthcolAbegintabularlName\Ducks\Lionsendtabular
 captionI would like the formulae here to be vertically at the middle of the two lines of text
 begintabularxlinewidthpdimexprcolA-2tabcolsepX>$c<$
 toprule
 Name & Description & multicolumn1cFormula\
 midrule
 &leavevmodellapmakebox[colA][l]DucksSomething which goes on two lines, something which goes on two lines &
 A=dfracBC\
 &leavevmodellapmakebox[colA][l]LionsSomething which goes on two lines, something which goes on two lines and on one more line and on one more line and on one more line &
 D=dfracEF\
 bottomrule
 endtabularx
endtable

%Only for show some coffin code:

NewCoffinCoffinA
NewCoffinCoffinB
NewCoffinCoffinC

SetHorizontalCoffinCoffinADucks
SetHorizontalCoffinCoffinC$D=dfracEF$

SetVerticalCoffinCoffinBdimexpr textwidth-CoffinWidthCoffinA-CoffinWidthCoffinC-6tabcolsepnoindent Something which goes on two lines, something which goes on two lines and on one more line and on one more line and on one more line

JoinCoffinsCoffinA[H,r]CoffinB[T,l](2tabcolsep,0pt)
JoinCoffinsCoffinA[vc,r]CoffinC[vc,l](2tabcolsep,0pt)
noindenthspace*tabcolsepTypesetCoffinCoffinA
enddocument

Like this:

Edit:

Position of columns contents are determined by row baseline, which (unfortunately) cannot be changed from column to column.

So far I don't see any other possibility than to use boxes either in the last or the first column, which align their baseline. A good candidate for your particular case is adjustbox:

documentclassarticle
usepackageamsmath
usepackagetabularx,booktabs
usepackagecaption

usepackageadjustbox

begindocument
begintable[htb]
 centering
 captionI would like the formulae here to be vertically at the middle of the two lines of text
 begintabularxlinewidthlXc
 toprule
 Name & Description & Formula \
 midrule
 Ducks & Something which goes on two lines, something which goes on two lines &
 adjustboxvalign=t$A=dfracBC$\
 Lions & Something which goes on two lines, something which goes on two lines &
 adjustboxvalign=t$D=dfracEF$\
 bottomrule
 endtabularx
endtable
enddocument

For more convenient writing of the table, you can define a new column type:

newcolumntypeE>beginadjustboxvalign=t$c<$endadjustbox

and then write the table (body) as:

 begintabularxlinewidthlXE
 toprule
 Name & Description & Formula \
 midrule
 Ducks & Something which goes on two lines, something which goes on two lines &
 A=dfracBC\
 Lions & Something which goes on two lines, something which goes on two lines &
 D=dfracEF\
 bottomrule
 endtabularx

If the text in cells of the middle column has arbitrary number of lines, or if the equation has only one line, the result is worse. In such case a possible solution is the use of multirow in the first column and manually adjust the number of columns which those cells spans, and in the second column set baseline in vertical middle:

documentclassarticle
usepackageamsmath
usepackagebooktabs, multirow, tabularx
usepackagecaption

usepackageadjustbox
usepackagelipsum

begindocument
begintable[htb]
 centering
 renewcommandtabularxcolumn[1]m#1 % <---
 captionI would like the formulae here to be vertically at the middle of the two lines of text
 begintabularxlinewidthlX>$c<$
 toprule
 Name & Description & Formula \
 midrule
multirow-6*Ducks % <--- manually adjusted
 & lipsum[66] & A=dfracBC\
 bottomrule
 endtabularx
endtable
enddocument

or use solution proposed in @Bernard's answer.

To have a complete picture of the possibilities, I include a table typeset with the help of cals. It is very easy to have such alignments using a calstable. I have removed left and right side bearing, which add code, not strictly necessary for creating OP’s table:

documentclassarticle
usepackagecals, caption, amsmath
usepackagelmodern

begindocument

begintable*
captionI would like the formulae here to be vertically at the middle of the two lines of text
begincalstable[c]
% Defining columns relative to each other and relative to the margins
colwidthsdimexpr(columnwidth)/6relax
 dimexpr(columnwidth)/6*4relax
 dimexpr(columnwidth)/6relax
 

% Set up the tabular
makeatletter
defcals@framers@width0.8pt % Outside frame rules, reduce if the rule is too heavy
defcals@bodyrs@width0.6pt
defcals@cs@width0pt % Inside rules, reduce if the rule is too heavy
defcals@rs@width0.4pt

defbbifxcals@borderBrelax % Botton border switch (off-on)
 defcals@borderB0pt
else letcals@borderBrelaxfi

deflpifdimcals@paddingL=0.0ptrelax % Left padding switch (off-on)
 cals@setpaddingAg
else setlengthcals@paddingL0ptfi

defrpifdimcals@paddingR=0.0ptrelax % Right padding switch (off-on)
 cals@setpaddingAg
else setlengthcals@paddingR0ptfi


% R1H1
theadsmall%
brow
 lpalignLcellNamelp
 alignLcellDescription
 rpalignRcellFormularp
erow
normalsize%

tfootlastrulestrut
%R2B1
brow
 lpbbalignLcellDuckslp
 alignLcellSomething which goes on two lines, something which goes on two lines
 rpalignRcellvfil$A=dfracBC$rp
erow
%R3B2
brow
 bblpalignLcellGnuslp
 alignLcellSomething which goes on two lines, something which goes on two lines and on one more line and on one more line and on one more line
 rpalignRcellvfil$D=dfracEF$rpbb
erow
%R4B3
brow
 lpalignLcellGnatslp
 alignLcellSomething which goes on two lines, something which goes on two lines and on one more line and on one more line and on one more line
 rpalignRcellvfil $D=dfracEF$rp
erow
makeatletter
endcalstablepar
endtable*
enddocument