Springboard is the launcher for all iPhone apps (well, at least ones that you can launch). When jailbreaking and theming came around, Summerboard was the first tool to manipulate how the Springboard looked. When 2.0 came around, Winterboard took over where Summerboard started. See the progression?
There will inevitably be an Autumnboard someday, mark my word.
SpringBoard is basically the name of the "desktop" of your iPhone or iPod Touch, just like "Explorer" is the name of the one for Windows-based machines, and just like how "Finder" is the name of the one for Mac OS X-based machines.
WinterBoard is just the name of the app that let's you change the way SpringBoard acts and looks.
To open app that you have downloaded (through the App Store or through Cydia/Installer), just find the app in SpringBoard and tap it and it should open (you should already know that...).