What kind of commands produce the "example" and "solution" headings in the picture

Question

I've tried using flushleft with bolds such as

\begin{flushleft}{\medium{\textbf{Example 1}}\end{flushleft}

but that makes a large space right after.

Notice the text is all the way to the left of the pdf, there is minimal space between it and the next paragraph, the the paragraph is indented.

Answer 1

Here's an example with thmtools to let you start.

\documentclass{article}
\usepackage{lipsum} % for dummy text

\usepackage{amsthm,thmtools}

\declaretheoremstyle[%
  spaceabove    = \topsep,
  spacebelow    = \topsep,
  bodyfont      = \itshape,
  headpunct     = ,
  postheadspace = 1em]%
{definition}

\declaretheoremstyle[%
  spaceabove    = \topsep,
  spacebelow    = \topsep,
  headpunct     = ,
  postheadspace = \newline,
  postheadhook  = {\hspace*{\parindent}}]%
{example}

\declaretheoremstyle[%
  spaceabove    = \topsep,
  spacebelow    = \topsep,
  headpunct     = ,
  postheadspace = \newline,
  postheadhook  = {\hspace*{\parindent}}]%
{solution}

\declaretheorem[style=definition,name=Definition]{defn}

\declaretheorem[style=example,numberlike=defn,name=Example]{ex}

\declaretheorem[style=solution,numbered=no,name=Solution]{sol}

\begin{document}
\begin{defn}
\lipsum*[2]
\end{defn}

\begin{ex}
\lipsum*[2]
\end{ex}

\begin{sol}
\lipsum*[2]
\end{sol}
\end{document}

Output:

and an example without theorems...

\documentclass{article}
\usepackage{lipsum} % for dummy text

\newcounter{defn}
\newcommand{\Def}{\refstepcounter{defn}\vspace*{\topsep}\noindent\textbf{Definition \thedefn}\hspace*{1em}}
\newcommand{\Ex}{\refstepcounter{defn}\vspace*{\topsep}\noindent\textbf{Example \thedefn}\par}
\newcommand{\Sol}{\vspace*{\topsep}\noindent\textbf{Solution}\par}

\begin{document}
\Def{\itshape\lipsum*[2]}

\Ex\lipsum*[2]

\Sol\lipsum*[2]
\end{document}

Output:

Answer 2

If it's an introduction to LaTeX, maybe the OP is looking to something manual and basic like

\noindent\textbf{Example}\par
\indent text text text

?

Answer 3

A simple code with ntheorem. I don't think the 1st paragraph indent is nice, so I didn't set it. It can be inserted with a small patch of the break and nonumberbreak theorem styles defined by ntheorem.

\documentclass{report}%]

\usepackage[utf8]{inputenc}%
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage[thmmarks, amsmath]{ntheorem}%

\theoremstyle{break}
\theoremheaderfont{\bfseries\upshape}
\theorembodyfont{\mdseries\hspace*{1em}}
\theoremseparator{\smallskip}
\newtheorem{example}{Example}[section]
\renewcommand\theexample{\arabic{example}}
\theoremstyle{nonumberbreak}
\newtheorem{solution}{Solution}


\begin{document}

\begin{example} \label{ex-1}
 Some example%
\end{example}

\begin{solution}
Some solution for example \ref{ex-1}. 
\end{solution}

\end{document}