Knuth, when designing the current version of TeX back in the early
1980s, could discern no "standard" way of expressing graphics in
documents. He reasoned that this state could not persist for ever,
but that it would be foolish for him to define TeX primitives that
allowed the import of graphical image definitions. He therefore
deferred the specification of the use of graphics to the writers of
DVI drivers; TeX documents would control the drivers by
means of
\special commands.
There is therefore a straightforward way for anyone to import graphics
into their document: read the specification of the \special
commands your driver uses, and 'just' code them. This is the
hair-shirt approach: it definitely works, but it's not for everyone.
Over the years, therefore, "graphics inclusion" packages have sprung up; most were designed for inclusion of Encapsulated PostScript graphics - which has become the lingua franca of graphics inclusion over the last decade or so.
Notable examples are the epsf package (distributed with dvips) and the psfig package. (Both of these packages were designed to work well with both Plain TeX and LaTeX 2.09; they are both still available.) All such packages were tied to a particular DVI driver (dvips, in the above two cases), but their code could be configured for others.
The obvious next step was to make the code configurable dynamically. The LaTeX standard graphics package and its derivatives made this step: it is strongly preferred for all current work. It can also be used (with the help of the miniltx "LaTeX emulator" and the graphicx.tex front-end) in documents written in Plain TeX.
The graphics package takes a variety of "driver options" - package options that select code to generate the commands appropriate to the DVI driver in use. In most cases, your (La)TeX distribution will provide a graphics.cfg file that will select the correct driver for what you're doing (for example, a distribution that provides both LaTeX and PDFLaTeX will usually provide a configuration file that determins whether PDFLaTeX is running, and selects the definitions for it if so).
The graphics package provides a toolkit of commands (insert
graphics, scale a box, rotate a box), which may be composed to provide
most facilities you need; the basic command, \includegraphics,
takes one optional argument, which specifies the bounding box of the
graphics to be included.
The graphicx package uses the facilities of of
graphics behind a rather more sophisticated command syntax
to provide a very powerful version of the \includegraphics
command. graphicx's version can combine scaling and
rotation, viewporting and clipping, and many other things. While this
is all a convenience (at some cost of syntax), it is also capable of
producing noticeably more efficient PostScript, and some of its
combinations are simply not possible with the graphics
package version.
The epsfig package provides the same facilities as
graphicx, but via a \psfig command (also known as
\epsfig), capable of emulating
the behaviour (if not the bugs) the old psfig package.
Epsfig also supplies homely support for former users of the
epsf package. However, there's a support issue: if you
declare you're using epsfig, any potential mailing list or
usenet helper has to clear out of the equation the possibility that
you're using "old" epsfig, so that support is slower
coming than it would otherwise be.
There is no rational reason to stick with the old packages, which have never been entirely satisfactory in the LaTeX context. (One irrational reason to leave them behind is that their replacement's name tends not to imply that it's exclusively related to PostScript graphics. The reasoning also excludes epsfig, of course.)
A wide variety of detailed techniques and tricks have been developed over the years, and Keith Reckdahl's epslatex outlines them in compendious detail: this highly recommendable document is available from CTAN. An invaluable review of the practicalities of exchanging graphics between sites, "Graphics for Inclusion in Electronic Documents" has been written by Ian Hutchinson; the document isn't on CTAN, but may also be browsed on the Web.
This question on the Web: http://www.tex.ac.uk/cgi-bin/texfaq2html?label=impgraph