Essentially, Arduino and AVR are different terms which cannot be compared. We will begin right from the basics. Let us begin by talking about the microcontroller as well as its development board. A microcontroller is a microcomputer which is built on a single chip.
You can build a lot of things from these microcontroller units ranging from a water level monitoring system to an etch-a-sketch wall clock. You can begin by utilizing the microcontroller but you will also need to build up the interface circuitry so as to connect the microcontroller to a personal computer to be able to load the program or even to run the motor. This is where a development board comes in handy. It will save you the time and effort of building the major part of the interface circuitry that will be needed. Development boards have the microcontroller chips soldered or plugged into them.
Now, let us understand what is an AVR. An AVR is a series of microcontroller processors just as an iPhone is a series of mobile phones. For example, ATmega16 is a particular microcontroller under the AVR series of microcontrollers.
Now to understand, what is an Arduino? An Arduino is series of development boards built mostly using the AVR microcontroller processors. For example, the model Arduino Uno R3 is one of the development board in the Arduino series. It is created for an AVR microcontroller, the ATmega 328.
What is an AVR development board? Any particular development board that makes use of an AVR microcontroller is an AVR development board. The Arduino Uno is one of those particular boards. Some of these boards may be a little more complex like the ones which provide better functionalities. But in reality, they all have an AVR microcontroller as part of their core. Hence, an Arduino board can be an AVR development board if it makes use of an AVR microcontroller.
The concept of a development board has been in existence ever since the electronics industry has existed. The manufacturers are very particular about providing these so as to show off their parts. But over the past few years Arduino has become much more than just a development board. Arduino has been able to build a completely new ecosystem around itself. There is also a lot of helping material available around Arduino, which is not the case for the other boards. Books, websites, Wikipedia pages, libraries, free manuals, shields, and also a lots of open source code material and the designs are what really sets Arduino apart from everything else available.
There are essentially two ways to be able to program an AVR microcontroller. One is to reprogram the entire microcontroller making use of an AVR programmer. The other method is using a bootloader which is pre-programmed onto that microcontroller which allows the microcontroller to re-program itself. An AVR programmer is also a lot more powerful and you can virtually do anything on the chip and the full 32K of the memory is available to you. It is much safer to be using a bootloader as you will not be able to mess around with the fuse settings on the microcontroller but you will be having just the 30K of memory available as the balance2K will be used by the bootloader. It may not sound a really big deal to us, but if you will be working on a major big project which needs a lot of flash space, you will very well be needing it.