Symbol for a set of integers in LaTeX

Some general comments and observations:

• Assuming you use pdfLaTeX to compile your document, then unless you either load a font that natively provides "blackboard bold" (aka "double-struck") uppercase letters or load some other package which loads a suitable math font, you will need to load the amsfonts package in order to access the \mathbb macro.

• Loading the amssymb package "works" too, since amssymb loads amsfonts automatically.

• The pdfLaTeX kernel does not provide commands named \N and \Z by default. (Aside: I have no idea why the website you provide a link to claims that one can use \N directly in a LaTeX document.) However, as is shown below, it's rather straightforward to create macros named \N and \Z which, in turn, execute \mathbb{N} and \mathbb{Z}, respectively. (Or, if you prefer, load the dsfont package and define \N, say, via \newcommand{\N}{\mathds{N}}.)

• If you can use either LuaLaTeX or XeLaTeX to compile your document, you may want to load the unicode-math package to get access to its \symbb macro. This lets you create double-struck characters not just for uppercase letters but for lowercase letters and numerals as well. An example: $\symbb{ABCabc123}$. A

The test program used to create the following screenshot employs pdfLaTeX and shows the symbols frequently used to denote the sets of integers ("Natürliche Zahlen" in German), whole numbers ("ganze Zahlen"), rational numbers, real numbers, and complex numbers.

\documentclass[border=1pt]{standalone} 
\usepackage{amsfonts} % for "\mathbb" macro
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}

\begin{document}
$\N \quad \Z \quad \Q \quad \R \quad \C$
\end{document}

As said by others, \mathbb is defined in amsfonts. For example,

is obtained with the following code.

\documentclass{article}
\usepackage{amsmath,amsfonts}
\begin{document}
\(\mathbb{Z}\)
\end{document}

However, I think it's worth here to mention the existence of the mathalpha package (see the documentation here) which allows to use many fonts with \mathbb. For example,

\documentclass{article}
\usepackage[bb=ams]{mathalpha}
\begin{document}
\(\mathbb{Z}\)
\end{document}

gives the same output than on the image given earlier. But other fonts could be used, for example

is obtained by replacing bb=ams with bb=boondox in the preceding code, and

is obtained with bb=pazo.

The other answers all say how to use legacy 8-bit fonts, which still work. If you’re using modern fonts with LuaLaTeX or XeLaTeX, it’s enough to load unicode-math. It also defines \BbbZ and lets you type in ℤ in your source.

You can also write \newcommand\Z{\mathbb Z} if you want that alias.