OSGeo is an umbrella organisation (Foundation) that supports many Open Source GIS Projects. Some of the more well known ones are: QGIS, GeoServer, and OpenLayers.
Being part of OSGeo gives a project some support, through assistance with governance, and potentially funding. But it also gives a project some legitimacy and assurance. Being part of OSGeo means the project has a team of core developers, a roadmap on where the project is going, and some form of governance.
You can be confident using Open Source GIS software because it has this support network through OSGeo.
OSGeo4W is a windows installer for Open Source GIS projects. The nature of Open Source means that many project/programs rely on eachother for features. A great example of this is GDAL. GDAL is used to some extent by practically all Open Source GIS projects to read and write data. But because Windows, being a closed platform, it has not developed a package manager like Unix based operating systems. So if you install GRASS and QGIS using their stand-alone installers, you end up with 2 installs of GDAL. Add 3 or 4 more Open Source installs and you can end up with a dozen installs of GDAL, in addition an install just for GDAL.
This is where OSGeo4W comes in. It can keep track of the shared requirements of Open Source GIS packages, so QGIS and GRASS can share a single install of GDAL. It also keeps track of versions so you can simply upgrade programs.