Equation with indices at end and to right side

Default positions of equation numbers and other tags at document class amsart are on the left side of pages. To move them on the right side, you need to add option reqno to document class options. This will move equation numbers and other tags to the right:

 \documentclass[11pt, reqno]{amsart}
 \usepackage{amsmath,,amssymb, array}
 \begin{document}

  Consider following system of equations:

  \begin{equation*}
  a_i x_i^2 + b_i = 0    \tag{$i=1,2,\ldots, m$}
  \end{equation*}
  \end{document}

Edit: In the case, when you like to have just this tag on the right side and equation numbers on the left, than the answer of Andrew Swann on question is handy. Its adaption to your case is:

\documentclass[11pt]{amsart}
\usepackage{array}
\usepackage{amsmath, amssymb}
\makeatletter
\newcommand{\leqnos}{\tagsleft@true\let\veqno\@@leqno}
\newcommand{\reqnos}{\tagsleft@false\let\veqno\@@eqno}
\leqnos
\makeatother

\begin{document}
Consider following system of equations:
\begingroup\reqnos
    \[
a_i x_i^2 + b_i = 0    \tag{$i=1,2,\ldots, m$}
    \]
\endgroup
and
    \begin{equation}
    c^2 = y^2 + b^2
    \end{equation}
\end{document}

which gives:

Another options in this case is use the answer of Werner of on question, which adaption to your case is:

\documentclass[11pt]{amsart}
\usepackage{array}
\usepackage{amsmath, amssymb}
\makeatletter
\newcommand{\leqnomode}{\tagsleft@true}
\newcommand{\reqnomode}{\tagsleft@false}
\makeatother

\begin{document}

\reqnomode
    \begin{gather*} % <--- works only with amsmath environments
a_i x_i^2 + b_i = 0    \tag{$i=1,2,\ldots, m$}
    \end{gather*}
\leqnomode
and
    \begin{equation}
    c^2 = y^2 + b^2
    \end{equation}
\end{document}

Result is the same as before.