Is Node.js mature enough to make entire web applications in it? I mean entirely in Node.js. I read somewhere that for some reason there should be (for example) nginx behind
Its absolutely mature enough. There are dozens of companies now that are using it in production for major features, and some that are even building their entire stacks with it.
If you want a well-rounded web framework, check out Express.
There are hundreds of modules that cover almost every type of feature you need, and they're mostly all easily accessible via npm (node package manager).
Over the summer I built a high-traffic Facebook app for a client using Node. Handles everything just fine.