It is a good idea to have commands such as \R for the real numbers and
other standard number sets. Traditionally these were typeset in bold.
Because, in the ordinary course of events, mathematicians do not have
access to bold chalk, they invented the special symbols that are now
often used for \R, \C, etc. These symbols are known
as "blackboard bold". Before insisting on using them, consider
whether going back to the old system of ordinary bold might not be
acceptable (it is certainly simpler).
A set of blackboard bold capitals is available in the AMS msbm fonts (msbm is available at a range of design sizes, with names such as msbm10). The pair of font families (the other is called msam) have a large number of mathematical symbols to supplement the ones in the standard TeX distribution, and are available in Type 1 format with most modern distributions. Support files for using the fonts, both under Plain TeX and LaTeX (packages amssymb and amsfonts), are available. The font shape is a rather austere sans, which many people don't like (though it captures the essence of quickly-chalked writing rather well).
The bbold family is set of blackboard bold fonts written in Metafont. This set offers blackboard bold forms of lower-case letters; the font source directory also contains sources for a LaTeX package that enables use of the fonts. The fonts are not available in Type 1 format.
The bbm family claims to provide 'blackboard' versions of most of the cm fonts ... including the bold and bold-extended series. Again, the fonts are designed in Metafont and are not available in Type 1 format. LaTeX macro support comes from a package by Torsten Hilbrich.
The doublestroke family comes in just roman and sans shapes, at a single weight, and is available both as Metafont sources and as Type 1; the font covers the uppercase latin letters, lowercase 'h' and 'k', and the digit '1'.
An alternative source of Type 1 fonts with blackboard bold characters may be found in the steadily increasing set of complete families, both commercial and free, that have been prepared for use with (La)TeX (see "choice of outline fonts"). Of the free sets, the txfonts and pxfonts families both come with replicas of msam and msbm (though, as noted elsewhere, there are other reasons not to use these fonts). The mathpazo family includes a "mathematically significant" choice of blackboard bold characters, and the fourier fonts contain blackboard bold upper-case letters, the digit '1', and lower-case 'k'.
The "lazy person's" blackboard bold macros:
\newcommand{\R}{{\sf R\hspace*{-0.9ex}%
\rule{0.15ex}{1.5ex}\hspace*{0.9ex}}}
\newcommand{\N}{{\sf N\hspace*{-1.0ex}%
\rule{0.15ex}{1.3ex}\hspace*{1.0ex}}}
\newcommand{\Q}{{\sf Q\hspace*{-1.1ex}%
\rule{0.15ex}{1.5ex}\hspace*{1.1ex}}}
\newcommand{\C}{{\sf C\hspace*{-0.9ex}%
\rule{0.15ex}{1.3ex}\hspace*{0.9ex}}}
are almost acceptable at normal size if the surrounding text is
cmr10. However, they are not part of a proper maths font,
and so do not work in sub- and superscripts. Moreover, the size and
position of the vertical bar can be affected by the font of the
surrounding text. As we've seen, there are plenty of alternatives:
don't try the macros, or anything similar using capital 'I' (which
looks even worse!).
This question on the Web: http://www.tex.ac.uk/cgi-bin/texfaq2html?label=numbersets