How can I change the settings of my table of contents, so that the sections and subsections are not indented as shown in the second picture. I would also like to center the title of the Contents. I am using the article class.

How can I change the settings of my table of contents, so that the sections and subsections are not indented as shown in the second picture. I would also like to center the title of the Contents. I am using the article class.

Using the tocloft package, all you have to do is to set the indents to the same values: 0em (so entries are flushed to the left margin) and 2em (space to typeset the unit number). Using \cfttoctitlefont and \cftaftertoctitle you can center the title; I also set tocdepth to 2 since subsubsections are not to be listed according to the image:
\documentclass{article}
\usepackage{tocloft}
\cftsetindents{section}{0em}{2em}
\cftsetindents{subsection}{0em}{2em}
\renewcommand\cfttoctitlefont{\hfill\Large\bfseries}
\renewcommand\cftaftertoctitle{\hfill\mbox{}}
\setcounter{tocdepth}{2}
\begin{document}
\tableofcontents
\clearpage
\section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
\section{Test section one}
\subsection{Test subsection}
\subsection{Test subsection}
\subsubsection{Test subsubsection}
\subsection{Test subsection}
\section{Test section two}
\subsection{Test subsection}
\subsection{Test subsection}
\subsubsection{Test subsubsection}
\subsection{Test subsection}
\end{document}

This is an approach which uses the »tocstyle« package with the tocflat option along with the standard style.
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage[tocflat]{tocstyle}
\usetocstyle{standard}
\usepackage{blindtext}
% Redefinition of ToC command to get centered heading
\makeatletter
\renewcommand\tableofcontents{%
\null\hfill\textbf{\Large\contentsname}\hfill\null\par
\@mkboth{\MakeUppercase\contentsname}{\MakeUppercase\contentsname}%
\@starttoc{toc}%
}
\makeatother
\begin{document}
\tableofcontents
\blinddocument
\end{document}
You can compare with the tocfullflat option for equidistant spaces between numbers and heading entries. More details can be found in the package manual.
N.B.: You will get a warning about the alpha status of the package. So far is has not caused any major problems for me.

Here's a solution using titletoc package; it defines the dottedcontents command which has syntax:
% \dottedcontents{<section>}[<left>]{<above-code>}
% {<label width>}{<leader width>}
I have also used the command
\renewcommand{\contentsname}{\centering Contents}
to center the table of contents title.

Here's the code:
% arara: pdflatex
% arara: pdflatex
\documentclass{article}
\usepackage{titletoc}
% \dottedcontents{<section>}[<left>]{<above-code>}
% {<label width>}{<leader width>}
\dottedcontents{section}[0em]{\bfseries}{2.9em}{1pc}
\dottedcontents{subsection}[0em]{}{3.3em}{1pc}
% center the toc heading
\renewcommand{\contentsname}{\centering Contents}
\begin{document}
\tableofcontents
\section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
\section{Test section one}
\subsection{Test subsection}
\subsection{Test subsection}
\subsection{Test subsection}
\section{Test section two}
\subsection{Test subsection}
\subsection{Test subsection}
\subsection{Test subsection}
\end{document}
For further reading, see How to modify the space between the numbers and text of sectioning titles in the table of contents?.
\documentclassyou're using. – Werner Feb 03 '14 at 20:31