I often find statements on Android forums like, "Nexus devices are perfect for developers." or "A Nexus device is almost impossible to actually 'brick'" (Is this actually true?). What is it that makes the Nexus devices so special? Is it because they have the stock Android, and so tinkering with Android in its purest form is possible?
It's just that I own a Nexus 4, and I don't even know why developers love it this much!
No carrier or manufacturer bloatware. If you want to test apps or OS changes and be sure you're not hitting some device-specific bug, a Nexus device makes it less likely that will happen.
They all use the same tool for flashing ROMs: fastboot. Not all third-party devices use it, and having to learn a new set of commands for every device you have is quite annoying.
Easy to unlock bootloader. Many third-party devices don't give you a means to unlock the bootloader (in order to flash your own ROM) without tricksy hacks. Sometimes the manufacturer does this; sometimes the carrier adds it to avoid you getting around their restrictions on tethering or something else. On Nexus devices, it's as easy as fastboot unlock-bootloader.
You can compile your own ROM for a Nexus device from the AOSP sources. If you want to hack on Android itself, or if you simply want a userdebug build in order to allow more debugging options for apps, it's as simple as downloading and compiling the source. Third-party ROMs don't usually have source available, so to build a custom ROM, you have to reverse-engineer the ROM and/or port AOSP to the device yourself (or wait for other modders to do it). This is why the T-shirts worn by engineers on Google's Nexus programme describe it as "pre-hacked Android".
Google puts a lot of work into keeping the logs (the ones you get from logcat) clean, to make development easier. They block code changes that add too much noise to the log, so that it stays useful for debugging. Other manufacturers aren't usually so careful: they don't see the log as something their customers will want, so they let it get noisier and noisier with driver debugging info nobody will want to look at. Clean logs make it much easier for app and OS developers to see and debug the behaviour of their code.
Nexus devices tend to get updates to new Android versions before third-party devices. If you're an app developer, and you want to make sure your app works with a new Android version as soon as possible, or you want to take advantage of new features and test on a real device (not an emulator), then you don't want to wait for a third-party manufacturer to get their finger out and port the new Android version to their hardware.
In addition to what Dan Hulme wrote, I'd like to add that nexus devices have stock firmware images for previous versions of Android, which can be easily installed. You can downgrade your Galaxy Nexus to ICS, or your Nexus S to Gingerbread. The Nexus 4 shipped with 4.2 but soon yours will have 4.4. You will be able to load older versions later on if that kind of testing is important to you.
I found this to be very valuable when testing with the Nexus S because it went from 2.3 to 4.1. You can even rely on cyanogenmod to extend the life of the phone as a test platform, if you trust that cyanogen is sufficiently close to stock for app compatibility purposes.