Once you have downloaded a module from this site, and have unzipped it, you will have two files: a license file, and a .lua file, which is the module itself (eg. columnmarks.lua). Textadept modules should be installed in a directory below your /USERHOME/ directory.
On Linux, the /USERHOME/ directory is usually /.textadept/, located below $HOME (typically /home/username/), eg. /home/username/.textadept/.
On Windows, the /USERHOME/ directory is usually \.textadept\, located below the USERHOME environment variable (typically C:\Users\username\), eg. C:\Users\username\.textadept\. Alternatively, Textadept may be run with the command-line argument <-u userdata> to specify a different /USERHOME/ directory, typically for a portable installation. For example, if Textadept is installed in \textadept\ on a flash drive in drive D:, it could be run as D:\textadept\textadept.exe -u D:\textadept\userdata to make /USERHOME/ become D:\textadept\userdata\.
Lexer modules should be installed in /USERHOME/lexers/, eg. /USERHOME/lexers/takecommand.lua.
Other modules should be installed in a directory below /USERHOME/modules/. This may be done in one of two ways: