Putting two images beside each other

Actually there are a number of ways of achieving what you are asking for.

---

Without Using Any Package

documentclass{article}

usepackage{graphicx}

begin{document}

begin{figure}[!tbp]
  centering
  begin{minipage}[b]{0.4textwidth}
    includegraphics[width=textwidth]{flower1.jpg}
    caption{Flower one.}
  end{minipage}
  hfill
  begin{minipage}[b]{0.4textwidth}
    includegraphics[width=textwidth]{flower2.jpg}
    caption{Flower two.}
  end{minipage}
end{figure}

end{document}

---

Using Packages

You can use either subfig or subcaption.

Using subfig

documentclass{article}

usepackage{graphicx}
usepackage{subfig}

begin{document}

begin{figure}[!tbp]
  centering
  subfloat[Flower one.]{includegraphics[width=0.4textwidth]{flower1.jpg}label{fig:f1}}
  hfill
  subfloat[Flower two.]{includegraphics[width=0.4textwidth]{flower2.jpg}label{fig:f2}}
  caption{My flowers.}
end{figure}

end{document}

Using subcaption

documentclass{article}

usepackage{graphicx}
usepackage{caption}
usepackage{subcaption}

begin{document}

begin{figure}[!tbp]
  begin{subfigure}[b]{0.4textwidth}
    includegraphics[width=textwidth]{flower1.jpg}
    caption{Flower one.}
    label{fig:f1}
  end{subfigure}
  hfill
  begin{subfigure}[b]{0.4textwidth}
    includegraphics[width=textwidth]{flower2.jpg}
    caption{Flower two.}
    label{fig:f2}
  end{subfigure}
  caption{My flowers.}
end{figure}

end{document}

---

Pros and Cons of the Approaches

1. It is actually difficult to call one method superior over the other. Which one you want to use will depend on the result you are expecting. So, see the results presented above and choose yourself.
2. The first one which uses the minipage environment is actually very simple. But as you can see the figures are number individually. If want to present a group of related figures, it may not be the one you are looking for.
3. The results from subfig and subcaption are very similar. Though each has its own way of usage. However, there are reports on subfig not working properly with hyperref. This question provides an excellent discussion on the comparative analysis on subcaption vs. subfig.

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Further Reading

In order to get a better understanding of the placement and width controlling issues, I strongly suggest the you go through the documentation of the above two packages (subfig and subcaption). These documentations contain some excellent hints and examples.

Also, for comprehending the solutions of related issues, these questions (A, B, C, D, E, F) are worth taking a look at.

The MWE package offers a nice solution:

documentclass{article}
usepackage{mwe}% or load ’graphicx’ and ’blindtext’ manually
begin{document}
blindtext
begin{figure}
includegraphics[width=.48linewidth]{example-image-a}hfill
includegraphics[width=.48linewidth]{example-image-b}
caption{MWE to demonstrate how to place to images side-by-side}
end{figure}
blindtext
end{document}

These solutions are too complicated IMHO, you don't need to install any new packages.

begin{figure}[h]
begin{tabular}{ll}
includegraphics[scale=1]{Figures/Race.png}
&
includegraphics[scale=0.4]{Figures/Bearing.png}
end{tabular}
caption{Left: Diagram of angular contact bearing cite{NBCBearing}. 
Right: Disassembled bearing}
label{Fig:Race}
end{figure}

Not an exact solution, but a workaround. I wasn't happy with how much blank space Latex allowed between the two images. So I copied both images into MS word (or Google Docs), arranged them side-by-side, and used the Snipping Tool to create a "single" picture. What's seen below is a single .png file which is much easier to insert.

The conflict with subfig and hyperref does exist: it's not just a rumor, which is why I prefer subcaption as I almost invariably need hyperref.

Also, begin{subfigure}[b]{0.475textwidth} should give a cleaner finish, still with plenty of space between the figures.

Place:

usepackage{graphicx}
usepackage{caption}
usepackage{subcaption} 

in the preamble.

I would recommend using the tikz package. this answers your question in the most basic form, with for instance:

documentclass{article}

usepackage{graphicx}
usepackage{tikz}

begin{document}

begin{tikzpicture}
    node [anchor=north west] (imgA) at (.05linewidth, .050linewidth){includegraphics[width=.5linewidth]{figA.pdf}};
    node [anchor=north west] (imgB) at (.05linewidth, .500linewidth){includegraphics[width=.linewidth]{figB.pdf}};
end{tikzpicture}

end{document}

One advantage is that you can place any number of figure, but also place easily letters to identify the figures:

documentclass{article}

usepackage{graphicx}
usepackage{tikz}

begin{document}

begin{tikzpicture}
    node [anchor=north west] (imgA) at (.05linewidth, .050linewidth){includegraphics[width=.5linewidth]{figA.pdf}};
    draw [anchor=north west] (0., .000linewidth) node {(A)};
    node [anchor=north west] (imgB) at (.05linewidth, .500linewidth){includegraphics[width=.linewidth]{figB.pdf}};
    draw [anchor=north west] (0., .500linewidth) node {(A)};
end{figure}

end{document}

An example of use can be found in this paper's LaTeX code, at page 4 of the PDF:

You may need to tune the position of each object manually to get the appropriate result.