I've read a lot of things about HTTP 2 (which is still in development), so I also heard about the server push feature, but I my head, this is not clear.
Does this server push feature mean that the server will be able to send a response to the client without the latter making a request? Just like a vanilla TCP connection? Or I'm missing the point?
The HTTP2 push mechanism is not a generic server push mechanism like websocket or server sent events.
It is designed for a specific optimisation of HTTP conversations. Specifically when a client asks for a resource (eg index.html) the server can guess that it is going to next ask for a bunch of associated resources (eg theme.css, jquery.js, logo.png, etc. etc.) Typically a webpage can have 10s of such associated requests.
With HTTP/1.1, the server had to wait until the client actually sends request for these associated resources, and then the client is limited by connections to only ask for approx 6 at a time. Thus it can take many round trips before all the associated resources that are needed by a webpage are actually sent.
With HTTP/2, the server can send in the response to the index.html GET push promises to tell the client that it is going to also send theme.css, jquery.js, logo.png, etc. as if the client had requested them. The client can then cancel those pushes or just wait for them to be sent without incurring the extra latency of multiple round trips.
Here is a demo of push with SPDY (the basis for HTTP2) with Jetty https://www.youtube.com/watch?v=4Ai_rrhM8gA . Here is a blog about the push API for HTTP2 and SPDY in jetty: https://webtide.com/http2-push-with-experimental-servlet-api/
Essentially your understanding is correct, however, there is a lot more to it.
The server will only be able to send a resource to the client after a request for an HTTP page has been made and the resources required by that page for it to render properly, i.e. images, JavaScript files, CSS etc, have been identified. The mechanism responsible for this is the server side framework. In Java, this will be Servlet 4 and possibly JSF.
A server can not just send any resource to the client when it feels like it. Only under the above circumstance will it occur and a client will always be able to reject the server request to push a resource.
The mechanism of HTTP/2 server push has been really well designed and to get to grips with it I recommend this overview of HTTP/2 and this in depth article diving into the internals of the HTTP/2 protocol.