I can easily create a "Web App" on my iPhone by going to the "arrow" menu and attaching a website to the home screen.
Is it possible to do this on OS X with, say, Gmail or YouTube or anything else?
There's no built-in OS X behaviour that exactly matches that of iOS, but you can use what's often referred to as a site-specific browser to turn sites into OS X apps.
Fluid lets you turn sites into single apps that appear like any other OS X app — in their own windows, with separate dock icons. Pretty useful for web apps like Gmail, but you can use for on pretty much any site. It's free, but for $4.99 you get some extra features.
Like Fluid, but specifically for Gmail. Since it's specific to Gmail, it's a bit more finely tuned, with toolbar buttons for the usual sorts of mail app actions. $24.95, with a 30 day free trial.
That will give you a standalone app you can run to get a browser for a specific site. But it's quite limited compared to solutions like Fluid or Mailplane — as the name suggests, it's more of a popup than a standalone app.
This is my personal favorite and I've been using it for a while. Yes, it is an "inactive" project, but it still works well. It has a simple, easy to use interface:
I put up a blog post on how to use Mozilla Prism, in which I go into more detail about the various features.
Two successors to Prism
Mozilla Labs Chromeless
Chromeless | Mozilla Labs comprises two 2011 blog posts: - Prism is now Chromeless - Webian Shell: A full screen web browser built on Chromeless
– and whilst Webian Shell may be of interest to readers, it's not an answer to the question.
To a 2012 question, Chromeless “inactive”?, Michael replied:
Prism – MozillaWiki refers to Chromeless and WebRunner.
In Safari, add the website you're viewing as a bookmark, either to your bookmark bar, or the bookmark menu.
You can create a browser chrome-free version of a page by turning it into a dashboard widget. ( full instructions here: http://www.iclarified.com/entry/index.php?enid=273).
Creating a Browser Widget on OSX
This maybe isn't exactly what you want but it's similar behavior.
Google Chrome Apps, and Chrome App Launcher
From an Internet Archive Wayback Machine copy of the page:
Natifier does precisely this, for any app, for free, open source.