After I posted, cgnieder pointed out the existence of the nameref package, as in
\documentclass[11pt]{article}
\usepackage{nameref}
\begin{document}
I can even refer to section \nameref{s-1} before it is defined.
\section{Background so Long} \label{s-1}
I want to print just a number for Section \ref{s-1}.
But here, I want to print section \nameref{s-1}.
\end{document}

With it, \nameref{<label>} gives the full name of the section. In this regard, what I had suggested is slightly different, in that the name one assigns to the textual reference may or may not be the same as the full section title. Here was my original answer, edited to accentuate this difference:
I introduce optional arguments to \label to assign a textual reference (which, importantly, need not match the section name), and then use the newly defined [s] option to \ref to invoke the "special" textual reference rather than the usual [numerical] response.
By writing it to the aux file, I can invoke the textual reference before it is even defined.
\documentclass[11pt]{article}
\let\svlabel\label
\let\svref\ref
\makeatletter
\renewcommand\label[2][\relax]{%
\svlabel{#2}%
\ifx\relax#1\else
\immediate\write\@mainaux{\string%
\expandafter\gdef\noexpand\csname custom#2\noexpand\endcsname{#1}}%
\fi
}
\makeatother
\renewcommand\ref[2][\relax]{%
\ifx s#1\csname custom#2\endcsname\else\svref{#2}\fi
}
\begin{document}
I can even refer to section \ref[s]{s-1} before it is defined.
\section{Background so Long} \label[Background]{s-1}
I want to print just a number for Section \ref{s-1}.
But here, I want to print section \ref[s]{s-1}.
\end{document}

If one only needed to reference "backwards", meaning after a label was defined, one can dispense with the aux file write to a simpler alternative:
\documentclass[11pt]{article}
\let\svlabel\label
\let\svref\ref
\renewcommand\label[2][\relax]{%
\svlabel{#2}%
\ifx\relax#1\else
\expandafter\gdef\csname custom#2\endcsname{#1}%
\fi
}
\renewcommand\ref[2][\relax]{%
\ifx s#1\csname custom#2\endcsname\else\svref{#2}\fi
}
\begin{document}
I cannot refer to section ``\ref[s]{s-1}'' before it is defined.
\section{Background so Long} \label[Background]{s-1}
I want to print just a number for Section \ref{s-1}.
But here, I want to print section \ref[s]{s-1}.
\end{document}

hyperref? – Werner Feb 09 '21 at 20:32backgroundis the title of the chapter. It's not clear from your question, otherwise you'd just typebackgroundand don't need to use\ref. Correct? If so,namerefis the package for your, and you can use\nameref{<label>}. – Werner Feb 09 '21 at 22:27