How to define a school that set the @school variable?

LaTeX defines author as

DeclareRobustCommand*author[1]{gdef@author{#1}}

You can define school and department in the same way:

DeclareRobustCommand*School[1]{gdef@School{#1}}
DeclareRobustCommand*Department[1]{gdef@Department{#1}}

@ may not occur in the name of a macro on user level, so you have either to enclose the definitions between makeatletter and makeatother (and put the commands into your preamble) or to put them into a file mydefs.sty and load this file in the preamble with usepackage{mydefs}.

Probably the easiest way to create a new maketitle command is to leave the old one as it is and use it in your definition.

DeclareRobustCommand*Author[1]{gdef@Author{#1}}
newcommandMaketitle{%
   author{@Author@School, @Department}%
   maketitle
}

Note that we need a new author command, as we need the original command to combine school and department with the author.

documentclass{article}
makeatletter
DeclareRobustCommand*School[1]{gdef@School{#1}}
DeclareRobustCommand*Department[1]{gdef@Department{#1}}
DeclareRobustCommand*Author[1]{gdef@Author{#1}}
newcommandMaketitle{%
  author{@Author@School, @Department}%
  maketitle
}
makeatother
begin{document}
title{My Article}
Author{It's me}
School{my school}
Department{my department}
Maketitle
end{document}