An array in an equation with curly braces in both sides

As @siracusa has already pointed out in a comment, the all directive should be removed. The only other significant change I recommend making to your code is adding & alignment points inside the aligned environment. (After all, why employ an algin environment if no alignment is performed?)

enter image description here

documentclass{article}
usepackage{amsmath} % for 'aligned' environment
begin{document}

begin{equation}
delta(x)equiv
left{
begin{aligned}
0,     quad&text{if $xne 0$}\
infty,quad&text{if $x=0$}
end{aligned}
right},,
qquadtext{with }
int_{-infty}^{+infty}! delta(x),dx=1.
end{equation}   
end{document}

You can use cases to simplify it a little bit and, of course, remove all as @siracusa said.

documentclass{article}
usepackage{amsmath}
begin{document}

begin{equation}
delta(x)equivleft.begin{cases}
0,      & text{if } x ne 0 \
infty, & text{if } x = 0 
end{cases}right}, quadtext{with } int_{-infty}^{+infty}delta(x); dx = 1.
end{equation}

end{document}

enter image description here

You can compare four ways to get the result.

The alignment with the original code happens essentially by chance and needs visual formatting (additions of quad and ;).

With Bmatrix the alignment is again by chance (the second column items have the same width).

With cases, adding left. and right., a bigger space after equiv is used.

The best, in my opinion, is to use left{ and right}, with a nested aligned so you can clearly state the desired left alignment of both columns.

In any case, it should be text{if $x=0$}, so no awkward explicit ; is needed (which is not a normal interword space). Also , should precede dx, not ;.

documentclass{article}
usepackage{amsmath}

begin{document}

subsubsection*{Original without texttt{stringall}}
begin{equation}
delta(x)equivleft{begin{aligned}
0,quad
text{if};xne 0\
infty,quad
text{if}; x=0
end{aligned}right}quad
text{with}quadint_{-infty}^{+infty}delta(x);dx=1.
end{equation}  

subsubsection*{With texttt{Bmatrix}}
begin{equation}
delta(x)equiv
begin{Bmatrix}
0,      & text{if $xne 0$} \
infty, & text{if $x=0$}
end{Bmatrix}
quadtext{with}quad
int_{-infty}^{+infty}delta(x),dx=1.
end{equation}

subsubsection*{With texttt{cases}}
begin{equation}
delta(x)equiv
left.
begin{cases}
0,      & text{if $xne 0$} \
infty, & text{if $x=0$}
end{cases}right}
quadtext{with}quad
int_{-infty}^{+infty}delta(x),dx=1.
end{equation}

subsubsection*{With texttt{aligned}}
begin{equation}
delta(x)equiv
left{
begin{aligned}
&0,      && text{if $xne 0$} \
&infty, && text{if $x=0$}
end{aligned}
right}
quadtext{with}quad
int_{-infty}^{+infty}delta(x),dx=1.
end{equation}

end{document}

enter image description here

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