Unity will no longer support Web Player plugin starting with version 5.4 which is due to release in March 2016.
Unity made the decision due to the lack of support for NPAPI plugins in modern browsers. Chrome and Edge do not support the technology already while Mozilla announced it would follow suit with Firefox.
Jonas Echterhoff posted on the announcement blog
Clearly, the web ecosystem is moving away from browser plugins and we are quickly approaching the point where no current browsers will still be able to run plugin content. Given this outlook, Unity is diverting resources into alternative web technologies and will begin the end-of-life process of the Unity Web Player plugin.
When Unity 5.4 releases, developers wanting to push their content to web will need to use the WebGL export that is currently in preview phase and doesn’t require a browser plugin.
However, WebGL has different security mandates and performance differences compared to the Web Player to which Unity wrote
However, it is important to understand that WebGL is a different platform from the Web Player and does not match the feature set or performance of the Web Player. We are working closely with browser vendors to make sure this gap becomes as narrow as possible, but there are some limitations which are defined by the platform – such as restrictions on the networking protocols you can use, which are mandated by security concerns.
The engine developer also acknowledged legacy titles that have been build with the Web Player plugin and are looking into options on to approach those titles to keep them alive.
Unity deeply understands the importance and historical relevance of Web Player powered games and keeping this back catalogue of games playable is something we care about. We have formed a working group to investigate alternative technical solutions and will update the community as we progress.