This was written in reply to someone on Google+ who converted from Windows to Linux, and still claimed, that he needed to download software from the Internet, which I assumed meant 'from the web', and not a distro-native package repository.
Downloading software on Linux is different from Windows.
Now, each major Linux distribution (distro) relies on their own software repository; each distro uses one of the two major package management systems, and package management software built around them.
Ubuntu is based on Debian. In Debian, the very basic package management system is dpkg, but actual work is done by APT (A Package Tool). Some other distros use RPM as their package management system.
Package management software is usually component- and command line-based, and typically has a graphical front-end.
So, Debian and Debian-based distros use dpkg, and it is relied on by the more automated APT, which can be operated with Synaptic, which is a graphical front-end to APT. Individual packages each have the .apt extension appended to their filenames.
For comparison, dpkg is a basic installer component with a purpose similar to Windows Installer. dpkg may check for dependencies and report those to the user, if there's a software package that is missing.
APT still uses dpkg, but greatly automates package management. Missing packages are downloaded either in console (via command line), or with Aptitude (a text-based graphical program), or with the fully-graphical Synaptic (not to be confused with touchpad pointing device maker Synaptics).
Synaptic can be used with Ubuntu, too, but Ubuntu has its own primary app store-like program called Ubuntu Software Centre (_Software Center_ in U.S. spelling). This is what you should use to download software for Ubuntu.
There can be two or three major software packages, that can be downloaded separately from the package manager. One of these is LibreOffice, because each new LO release offers more and better features; and the other is Wine (their site is winehq), which can run applications made for Windows. Wine is not compatible with all Windows programs.
There is a possibility, that Ubuntu builds/backports versions of newer software to its LTS release, but I have not confirmed it, because I don't have experience with Ubuntu.
For most things, Linux already has a large number of functionally similar counterparts to Windows programs — with the exception of some specialized software with unique functionality that does not have a Linux-based alternative.