What is Unity?
First, in a picture, Unity is this (big image):
In short, Unity is an integrated game engine and editing environment. It's closest direct analog is probably the Unreal Development Kit, though Unity is much more flexible and has much more enticing licensing terms for indie developers. The engine itself isn't quite up to Unreal 3 standards, but it's damn close, and is more than passable for indie titles.
Unity comes in two forms. The basic Unity package is free to download and use. You can even build and release your game using the free package, provided it's not a commercial endeavor. If you want to release commercially, or use some of the Pro level features, there is a $1500 USD license fee. Pro does support several features that the free level does not, but they are not "game breaking". You can still create a full featured game with just the free package, you just lose some of the flexibility. For a complete version break down, see here:
There are also license add ons you can purchase for various platforms. The real beauty of Unity is that it supports basically every gaming platform you could possibly develop for. Obviously to develop for the consoles, you need to be in the registered developer program for that console and have a dev kit. For most indie developers, these are just pipe dream features, but it's nice to know the flexibility is there.
Why would I use Unity instead of X?
This depends entirely on what X is, but I have a few answers from various other packages I've used.
Unreal Development Kit
: This is a neat tool, but it ultimately greatly limits your flexibility to go commercial with your idea without Epic's direct involvement. The chances of an indie developer being able to "step up" to a full UE3 license are slim to none. If you really don't give a lick about ever taking your game commercial, I doubt Unity offers much this doesn't (although I think it's easier to start from "scratch" with Unity, as the UDK is still completely tooled to be an FPS system and that has to be stripped away).
: XNA is really a lower level tool than Unity. It doesn't provide the editor environment that Unity does. It's also not a complete engine, so much as a set of tools to interact with the XNA framework. If you are dead set on releasing your game on XBox, this is probably still your best bet unless you think you can get in the dev kit program. In addition, you lock yourself to Windows and XBox with this path.
: This is sort of the middle ground between XNA and UDK/Unity. No editing environment, and no game framework to speak of, but the 3D engine is very complete. Supports the major desktop platforms (Windows, OSX, Linux), but doesn't easily support the consoles. You will spend a lot of time writing a game engine and tools around this 3D engine, but if you just want a high performance 3D engine this is a good choice.
What are some basics I need to know?
Games in Unity are made up of three things: GameObjects, components and scripts. Scripts are actually a type of component, but we want to make a distinction between them. In addition, there are scenes, which are essentially the world that your game objects make up.
GameObjects in Unity are the building blocks of your game. They make up the physical and non-physical world objects in your various scenes. In their blueprint form, these are known as Prefabs and are displayed in your Project view. In the game world, they are full GameObjects and are displayed in your Hierarchy view.
Moving between the two states is seamless. You can build up a detailed game object from a series of prefabs, then drag that game object in to the Projects view and seamlessly turn it in to a prefab.
Game objects are made up of components. Components can be everything from an asset, such as a model or a texture, to a script. Components are how behavior is added to game objects. Adding a component to an object is as easy as dragging the component from your Project view and dropping it on the object in the Hierarchy you want your component to link to.
Unity contains tons of built in components. Everything from particle emitters to collision primitives.
Scenes are the sandboxes your game takes place in. In the simplest case, you can think of them as your levels. In reality, they are just containers of game objects. As such, they could have absolutely nothing graphical in them and still be a valid scene. A scene could just be a ton of scripts, or a camera pointing at a billboard with a 2D image, or just a GUI overlay. Things like your main menu or your loading screen will likely be represented as scenes.
Where can I get more info?
The best place to start is here: http://unity3d.com/unity/
You can download Unity here: https://store.unity3d.com/shop/
(don't worry, the free package just goes through the store for it's download)
You can start reading documentation here: http://unity3d.com/support/documentation/Manual/index.html
You can find various other resources, including some starter assets, here: http://unity3d.com/support/resources/
The helpful Unity community is here: http://forum.unity3d.com/
In addition, Unity has it's own StackOverflow-like system: http://answers.unity3d.com/