New installation – TeX editor says: "Error: Could not start the command"

(Also check out the MiKTeX tag wiki I just wrote up!)

The hardest part about getting started with LaTeX probably is learning what the different components you need are. Texmaker is an editor, but what actually compiles your source code and creates the document is within a (La)TeX distribution, the most popular of which are MiKTeX (Windows only) and TeX Live (Windows, Linux, OS X (as MacTeX)). You can read about some differences between the two at What are the advantages of TeX Live over MiKTeX?. I've always been using MiKTeX, which seems like a solid choice for a beginner to me. I am, however, considering trying out TeX Live as well.

If you decide in favor of MiKTeX, make sure to install the user mode (you're asked about that during the installation process), not in admin mode. It has nothing to do with multi-account configurations, but rather something network-y related. You can read about it at Difference between administrative and user mode of MiKTeX. But believe me, the user mode will make your life much easier. I wish I would've done it right away. (On another question that might come up before downloading MiKTeX: What is the difference between the basic MiKTeX and the complete MiKTeX?)

Once you installed a distribution, you might have to reinstall Texmaker if you want to avoid having to (find out and) type in all the paths yourself.

As introductory reading for LaTeX, I recommend The Not So Short Introduction to LaTeX2ε, which also has a section about the different parts of a working LaTeX system, if I remember correctly.