How do I write in Georgian with PDFLaTeX?

There are a few options to use Georgian in LaTeX. A simple one is to use a font that supports the Georgian alphabet, such as DejaVu Serif, and compile with XeLaTeX or LuaLaTeX. Tested in Ubuntu Linux:

\documentclass{article}
\usepackage{fontspec}
\setmainfont{DejaVu Serif}
\begin{document}
Some fonts support Georgian: კატები ყველგანაა
\end{document}

Result:

Another possibility is the mxedruli/xucuri package, for pdfLaTeX. For this you write the Georgian text using character codes described in the manual. The installation of the package is very straightforward using tlmgr or packages from your distribution package manager (such as apt).

MWE:

\documentclass[12pt]{article}
\usepackage{mxedruli}
\begin{document}
\begin{mxedr}
.gmertsa +semvedre, nutu .kvla damxsnas soplisa+sromasa,\\
cecxls, .cqalsa da mi.casa, haerta tana +sromasa;\\
momcnes prteni da a.gvprinde, mivhxvde mas +cemsandomasa,\\
d.gisit da .gamit vhxedvide mzisa elvata .krtomasa.
\end{mxedr}
\end{document}

Result:

The packages that you found are also useful, but more difficult to install - so maybe one of these two options are sufficient for your needs.

The accepted answer is good, but doesn’t say how to install this package.

First, if you can possibly run LuaLaTeX or XeLaTeX, and use Unicode fonts instead, you should. It’s much simpler and works much better. Otherwise, read on.

If your distro has packages that install these files for you, you should use those. On Debian or Ubuntu, you can look for an OS package with apt-file.

If you run MikTeX, you can use its installer to install both georgian and babel-georgian. If you run TeX Live, which is what most Linux distros use, you might be able to install babel-georgian through tlmgr, although Debian-based distributions disable this and make you manually install that too.

There is neither a TeX Live, Debian nor Ubuntu package containing the georgian files on CTAN (as of 2019). To install that, download the package contents from CTAN and unpack them under your $TEXMFLOCAL directory (typically /usr/share/texlive/texmf-local/). See the installation notes.

Whenever you manually install packages to the local texmf directory, you must run texhash and updmap-sys --enable Map=mapfile.map.

You could instead try installing to your home directory, under ~/texmf, but I do not recommend that: You would need to run updmap to create a font map under your home directory, which will break any updates to your system font map.