Based on my understanding, putting a table for everyone’s reference. I will not delve into pre-2001 history.
|Version||Assembly Version||Licensed Product – Better/Higher Product||Base Platform, Product with Limited Features, Free Version|
|2001||x||SharePoint Portal Server (SPS) 2001||SharePoint Team Services (STS)|
|2003||v11||SharePoint Portal Sever (SPS) 2003, Microsoft Content Management Server (M CMS) 2002||Windows SharePoint Services(WSS) 2.0|
|2007||v12||Microsoft Office SharePoint Server (MOSS) 2007||Windows SharePoint Services(WSS) 3.0|
|2010||v14||SharePoint Server 2010||SharePoint Foundation 2010|
|2013||v15||SharePoint Server 2013||SharePoint Foundation 2013|
|2016||v16||SharePoint Server 2016||NA**|
* v13 skipped due to 13 being considered unlucky number by many customers.
** SharePoint Foundation is no longer available with SharePoint 2016
Portal from SPS 2003 was dropped as it lead people to believe that it can be used
only for Portal development and nothing else.
Office from MOSS 2007 was dropped as it lead some people to believe that it is an Office product like Word and Excel.
Hope Microsoft has got the name right this time with SharePoint Server 2010.
Update: No change in name with the release of SharePoint Server 2013 and 2016
The Microsoft cloud versions of SharePoint were called BPOS (Standard, Dedicated) in past.
Now they are referred as Office 365 SharePoint Online, or simply SharePoint Online. Unlike the On Premise version, SharePoint Online does not use a year in its name. SharePoint Online is an evergreen platform i..e. its updated frequently and users get to use the latest version of SharePoint
Still confused with version of SharePoint Online? Ok, let me take an example. Do you know which version of Facebook do you use? Or which version of Gmail do you use? You don't know, right. Same is true with SharePoint Online. Use SharePoint Online without worrying too much about the version.
There are some minor variations though, SharePoint Online Dedicated / Dedicated vNext and SharePoint Online Multi Tenant(MT). And you can turn on or off First Release features for our Office 365 tenant. First Release tenants and its users usually get the new features in beta or preview stage, before other production tenants.