The PupGet manager installs pupgets, and you just click on a DotPup.
But those are just the packages that people have created specifically for Puppy.

MU has made a debian installer and a gslapt package, which will attempt to install software from the Debian and Slackware repositories, respectively. Those packages may or may not work immediately in Puppy.

You can also download software from the web, in which case it will come in one of two formats: binary or source. A binary package just needs to be installed. There are many package formats, but the most common are .rpm, .deb, .tar.gz, and .tgz. The last two are the same, just different name systems. There are undeb and unrpm packages available on the wiki, and Puppy can handle tar.gz already (tar -zxf package.tar.gz). Once they are extracted, you usually put them in the places they come out in. For example, they will usually have a usr directory, so you stick that in / (so that usr and /usr merge). Always check readme's when available.

Source packages come in the same formats as above, but they contain the source code, which needs to be compiled. That's a whole 'nother discussion, but the basics are to open a terminal in the directory and run

First you need to install a devx_xxx.sfs module from http://www.puppyos.com/test/∞ (where xxx is your version of Puppy).