Arduino is a board, convenient for learning/prototyping/research, that is based on a microcontroller chip such as Atmega328. It not only provides supporting circuitry like voltage regulation etc., it also breaks out the pins of the microcontroller into comfortable input/output slots into which you can insert jumper wires and connect to external stuff on a breadboard, and also the Arduino board provides on-board USB communication so you can program it with your computer (for which the Arduino IDE/software/"C-C++ language" is very convenient and user-friendly).
For me, the phases whether over time with my learning evolution or over the course of a single project are like this:
1) Prepare/Test a circuit idea by connecting stuff (LEDs, buttons, analog-to-digital-converters, sensors, etc.) on a breadboard and the Arduino board, and coding the Atmega chip using Arduino.
2) Subtract the Arduino board, and make a "standalone Arduino", i.e., an Atmega chip by itself without anything excessive, on a breadboard (like in the URL below), and connect your whole circuit (LEDs, buttons, sensors, etc.) on the breadboard to the Atmega chip, and add USB communication if you want.
Arduino on breadboard tutorial: http://arduino.cc/en/Tutorial/ArduinoToBreadboard
3) Subtract the standalone Arduino, and once the circuit has been tested, solder the whole circuit using the chips and wires onto a stripboard/Veroboard, for compactness and durability.
4) Subtract the stripboard/Veroboard circuit, and prepare a schematic using software such as Eagle, generate Gerber files from it, and send them off to PCB manufacturer, and you can have a fully ready, compact and reliable circuit in the form of a PCB.
Any of the above four would be fully-functioning for whatever task one intends but differ in the final shape they take... To me, they are just different stages of prototyping and have different advantages/disadvantages.