The official information on Unity3D states that Unity is "More than an engine", and goes on to describe Unity as an "ecosystem of tools and services". This sounds great as a marketing soundbites, but what does that actually mean? Lets break it down in to smaller chunks:
We've all heard of game engines before; CryEngine, Unreal Engine, Source Engine, Unity Engine, Avalanche Engine... the list could go on for a while, but what exactly is this "engine" people keep talking about? The term "engine" comes from the metaphor of likening game design to a cars construction. The look of the car would be the art in the game, the steering wheel is the input, and so on. The "engine" is the part of the whole that makes it move, allows the gears to work, steering to function etc. and is generally regarded as the core of what a car is. Without an engine, a car can't do anything, much as a game without an engine can't do anything.
In terms of the Unity Game Engine, it is a high-level, multi-platform, pre-existing framework which allows a developer to create the game at hand (note: I didn't specify a coder here, as Unity can empower non-coders to achieve much more!). It is pre-existing code; classes, functions, co-routines, variables and more which can be used and manipulated in the right way to produce the game required.
Without an engine, everything would have to be re-written each time you wanted to create a game, which is massively time consuming and inefficient, and requires a far lower-level understanding of coding in order to achieve very basic things. This isn't impossible to do, but the skills required are far greater and more specific. At the end of such a task, you would have a game engine, which you can (ideally) re-use for other games.
When you download Unity to start making your game, you are not downloading the Unity Engine as a standalone entity to work on to "build you car around". What you are downloading is the Unity Editor, which really is where the greater power lies in Unity.
The Unity Editor is generally what is though of when people talk about Unity or Unity3D (the company that owns Unity3D is Unity Technologies). It is this editor that gives developers from all schools the ability to produce their own games, which use the Unity Engine. The editor is where all the tools are integrated in to one; are unified, hence the "unity" name. There are many facets to the Unity Editor including the Scene Object Hierarchy, the Inspector, The Project viewer, Game and Scene windows, and more, which allow the developer to work on the current project at a very high level, without losing out on the power that is lying beneath. It is the key in the Car metaphor. It opens the door, but it also starts the engine!
In the next tutorial, we'll look at how to download and install the latest version of Unity from scratch.