Installing Eggdrop is a relatively simple process provided your shell has the required tools for successful compilation. On most commercial shell accounts which allow Eggdrop bots you won't have any problems with installation, but on some private boxes or a shell on your ISP you may experience errors during compilation.
Below is a step by step guide to the installation process. These instructions apply to 1.6 bots. It assumes you will be installing eggdrop1.6.21.tar.gz, so just change the numbers if you are installing another version (in the 1.6 series, that is - installation of 1.4 and older versions varies slightly from 1.6).
1) Put the Eggdrop source on your shell using one of the specified download locations, either by downloading the eggdrop1.6.21.tar.gz file to your system then uploading it to the shell via FTP (recommended), or downloading it directly to the shell using the shell's FTP client. You don't need to put the .tar.gz file in its own directory (it'll be done automatically in the next step).
2) Telnet to the shell (if you haven't already), and type tar zxvf eggdrop1.6.21.tar.gz (if this doesn't work, try gunzip eggdrop1.6.21.tar.gz then tar xvf eggdrop1.6.21.tar). This will extract the Eggdrop source into its installation directory, named 'eggdrop1.6.21'.
3) Type cd eggdrop1.6.21 to switch to the directory the Eggdrop source was extracted to.
4) Type ./configure (that's a period followed by a slash followed by the word 'configure'). This makes sure the shell has all the right tools for compiling Eggdrop, and helps Eggdrop figure out how to compile on the shell.
5) When configure is done, type make config. This sets up which modules are to be compiled. For a more efficient installation, you can use make iconfig to select the modules to compile, but if you're not sure just use make config.
6) Type make. This compiles the Eggdrop. The process takes around two minutes or less on fast systems, longer on slow systems.
7) Type make install DEST=~/botdir. This will install Eggdrop into a directory named 'botdir'. You can change 'botdir' to anything you like.
Note that in some cases you may need to specify the full path, e.g. make install DEST=/home/cooldude/botdir - using the ~ character in make install won't always work. You can get the full path by typing pwd.
8) Switch to the root of your directory using cd ~ then type chmod 700 <botdir> (where <botdir> is the directory you installed the bot to). This is important to keep the contents your bot directory hidden from prying eyes.
9) You can safely delete the installation directory named 'eggdrop1.6.21' (to do this, type cd ~ then rm -rf eggdrop1.6.21) that was created previously, although some people may find it handy to keep that directory for performing additional or future installations of the same version without recompiling.
That's it! Eggdrop is now installed into its own directory on the shell. It's time to edit the configuration files to make Eggdrop work the way you want it to.