In the Linux world installing and using Latex is generally very simple. You need only to install a Latex IDE: Kile (if you use KDE) or Latexila (if you use Gnome) or whatever Latex IDE you prefer. The package manager of your distro does the hard work for you, installing all the needed Latex dependencies. I have never used Latex on Windows, but I suppose that projects like miktex should provide an easy installation too. An easy installation doesn’t mean a small installation. Latex is very powerful but it’s also very big and you’ll probably need to install a few hundred megabytes of software (depending on the optional latex packages). If you have a pc with limited disk space (e.g. a SSD-powered laptop) this can be a problem. An easy installation also doesn’t mean that you won’t have problems. You might want to use a Latex package that won’t work or that is not in your distro repository.
These are just two concrete motivations for using an online Latex editor. Yes, you write Latex code in your browser and you download the resulting pdf from your browser. Done. You don’t need to install any software on your pc. There are also many other motivations. You might be a Latex beginner and you want only to give it a try. You might be a Latex expert and you want to use a real-time collaborative editor with your colleagues. You might want to edit your projects from many different computers without install Latex on each of them. You can do many cool things with these tools. For example I absolutely love the large amount of free available latex templates (CV, resumes, etc.). You choose a template and you edit it, nothing more. Instead if you wish to use a latex template on your local latex installation, sometimes this can be a real pain.
Let’s see where you can find these editors. The two main competitors seem to be Sharelatex and Writelatex. They are quite similar but there are differences, of course. For example, sharelatex is open source, while writelatex is not. Both of them offer free plans, but also paid plans with advanced features like dropbox syncronization and revision control. The following table shows a quick comparison between features which you might be looking for.
As you can see, the choice depends on what you are more interested in. If you are a Latex beginner, probably you should consider the auto-completion of the Latex commands and then sharelatex. Instead if you are a Latex expert and you need free real-time editing with more than one collaborator, writelatex is the right choice for you.
What about privacy? Well, they are cloud services. Although they both offer private projects, you probably can’t trust them for the protection of really sensitive data. If you really need an online Latex editor over a trusted web storage, you can host an editor on your own server. Sharelatex can be freely self-hosted. Another open source alternative is FlyLatex, even though I don’t know much about it.