What is CyanogenMod and how is it different from Android?

I have been listening a lot of chatter about CyanogenMod and it only seems to be growing louder by the day. But what exactly is CyanogenMod and how is it different from Android? If CyanogenMod is a fork of Android then how can it (aim to) become third major mobile ecosystem. Aren't they both same?

Also, how safe is CyanogenMod, in the sense, can I trust it with my Contacts, Google, Facebook, Twitter and social account information?

Some time back I read an interesting article on Ars Technica - Google’s iron grip on Android: Controlling open source by any means necessary - which details out how Google is creating closed source equivalents of Android Open Source Project (AOSP) apps for e.g. Search, Keyboard and so on. If that's the case will Google's proprietary apps be available on CyanogenMod?

The same article details out how Amazon's fork of Android is not Google approved and because of that it faces many issues. Is CyanogenMod Google approved?

Answer

CyanogenMod versus Android

To ask how CyanogenMod is different from Android is comparable to ask How is a cat different from an animal. CyanogenMod is an Android custom-rom. See:

Quoting the latter, which quotes from the CyanogenMod website:

CyanogenMod is an aftermarket firmware for a number of cell phones based on the open-source Android operating system. It offers features not found in the official Android based firmwares of vendors of these cell phones.

As you're asking about Ecosystems: CyanogenMod meanwhile is a company. There might be multiple Ecosystems using the same resources. As will be shown below:

Google versus Android versus CyanogenMod

Android is not Google, and Google is not Android. Behind Android stands the AOSP team. So Android is Open source, which is why other groups like CyanogenMod can use it. The "iron grip" of your question doesn't go to Android itself, it's rather the additions Google puts on top, the so-called Google Apps (see: google-apps and its tag-wiki). Those apps are not open-source, but closed-source. CyanogenMod started shipping their own counter-parts for several of the Google-Apps (and features), and here you've got your ecosystem: Apps and Services.

CyanogenMod is not permitted to ship the Google-Apps with their ROMs, you have to install them manually (if you want them), or leave them out. There are many substitutes available, so you can use Android without Google:

Can I trust X with my data?

Google earns its money with advertisements. If you can trust them with your data, IMHO you can trust CyanogenMod even more. If you don't want to trust either of them, there are alternatives such as OwnCloud, Funambol, and more, which you can use with any ROM, CyanogenMod included – setting up your own cloud, as one of the names suggests.

Concerning the google-play-store and "where to get my apps from": As with the other google-apps, custom ROMs are not permitted to bundle them with their distributions. But there are a lot of alternative and open-source markets, like F-Droid etc. (see our alternative-markets tag-wiki). There have been rumours about a CyanogenMod app market, but I must have missed that taking off.