The Nexus 4 is Google's latest release in the Nexus line of devices. These are a series of phones and tablets that run on Android and run the latest and purest version of Android developed by Google at the time of their release. Device manufacturers generally modify Android with user interface tweaks and custom apps to try and "brand" themselves as a hardware company. While manufacturers may see that as a good deal, it may not be so great for Google or for consumers who have to wait for a long time for Android updates on other devices.
Nexus products show users that pure Android is sexy too. Nexus products are made by different manufacturers. In this case, the Nexus 4 is made by LG.
The Nexus 4 is so named because of its 4 inch diagonal size, and because it is the 4th branded Nexus phone.(It's actually 4.7 inches.) It comes in an 8 and 16 gig version, but neither version has LTE 4G, although it does support HSPA+, which is T-Mobile's slightly faster than 3G, slower than LTE 4G network. The phones sell in the US for $300 and $350 unlocked and without a contract. That's a really good price, even without the LTE. The phones, as of this writing, will run on GSM networks only, which means they won't work on Sprint or Verizon. They are great as international phones for those who like to visit Europe.
The Nexus 4 is also really hard to keep in stock. Google has sold out several times in the US, and as of this writing they're completely out. The UK release also sold out within hours. Google may have scrapped plans to release the phone in a few other markets in order to restock the online store for hungry US customers, and more phones should be headed to T-Mobile stores in the near future. (As I'm writing this, it's a week before Black Friday.)
What's so great about this phone?
Aside from the reasonable price for an unlocked phone and the access to more frequent updates, the Nexus 4 takes 360 degree panorama pictures in what Google calls photo spheres. Google+ already has the tools to view these new uploaded photo spheres. The phone also comes with improved photo editing tools built in.
You don't have to plug it in to charge it. The Nexus 4 is already compatible with inductive charging mats, but you'll have to buy your own. It doesn't ship with one. If you have one, however, that means you can avoid the late night forgetting to plug in a phone issue. Just set it on the mat, and it will charge. However, don't expect to carry around spare batteries for day trips, because those induction cables mean the battery in this one isn't easily replaceable.
"Android Beam." Nexus phones (other than the Nexus One) support NFC, and now they support tapping phones together with other Android users and instantly sharing whatever is on the screen. The NFC also means you've got access to Google Wallet, which is Google's phone-instead-of-credit-card payment system. It's available in limited locations, but where it's offered, you can tap your phone and then enter a pin code to make a payment rather than swiping your credit card.
Gesture typing - this is a feature available on other phones through Swype, but now it's part of the official Android release. You trace your fingers along the keys, and predictive software guesses what you're trying to type. Sometimes this works better than others - just try to have a keyboard predict "Marziah Karch" for example, but when it works it's actually much faster than tapping out keys individually with two thumbs.
And to back it all up, this LG phone comes with a quad core processor and Corning's Gorilla Glass 2 to protect the screen. Anecdotal testimony is that this phone seems faster even than some LTE phones, likely because switching between 3G and 4G networks as you move around takes a toll on the phone's performance.
The Nexus 4 has been hard to find in stock. In the US, the phone has been selling out within minutes of the release. I'm sure this will change in 2013, but as I am writing this, I've seen the store sell out within minutes three times. (I was one of those customers trying to purchase a phone.)
This is all pretty ironic considering the first Nexus branded phone, the Nexus One, was considered a commercial flop. It was a fantastic phone, but customers weren't wiling to pay for a phone without a data plan, and they were unsure of this whole idea of buying a phone directly on the Web from Google instead of buying it in a brick and mortar store. Google seems to have succeeded in doing exactly what they set out to do and directly marketed their consumer electronics to customers over the Web.