Drivers
Unlike Windows, most hardware is natively supported in Ubuntu and does not require separate drivers to work. Generally speaking, if it works don't worry about a driver not being installed. Please note: there is a bug so that your graphics card will be displayed as Unknown in System Settings even if the driver is correctly installed and working--you can safely ignore this (or see this Q&A for the workaround).
If however something isn't working:
Ubuntu uses a graphical user interface called jockey or "Additional Drivers" to manage and install hardware drivers that are not natively supported by Ubuntu.
To open it hit Alt+F2 and type jockey-gtk.
A window will then open that will allow you to select additional drivers for sound, video, wireless etc. Here's a screenshot of this window, I already have several drivers enabled.
If something still doesn't work and you don't see a relevant driver in Jockey, please post a specific question including as much information about your hardware and what you have tried!
Applications
New applications on the other hand--as opposed to drivers--will have to be installed manually. However, unlike Windows, Ubuntu manages software centrally using repositories--in short, you don't have to go find/download software yourself. Ubuntu Software Center is one option for installing software (see this question) but you can also install from the command line usingsudo apt-get install software-name (see this question for more information on finding/installing software from the command line). You can also download Debian packages (.deb) from the internet and manually install them (or compile software yourself from source code if you are feeling ambitious)--but it's much safer and more convenient to install from the repositories. Welcome to Ubuntu!
