An Arduino is basically a Microcontroller on a handy PCB with built-in programming interface. It also has a simple programming environment to get you up and running quickly.
The chip on the board can be thought of as the "brain" of the system. It's what you program and it's where you upload your software to.
Is one Arduino device required for each Arduino project?
Yes and no. You can't "remove" your Arduino from your project and expect it to still work - it'd be like beheading someone - they cease to function without their brain. However you can replicate the Arduino in your system fairly easily so you don't have to embed the whole Arduino into it. You can get a blank chip the same as is on the Arduino reasonably cheaply and program the bootloader (the serial interface for programming it) into the chip using your existing Arduino, then build a new circuit around it. More information here:
Should I have to buy more then one device for each project?
That depends if your project is complex enough to warrant more than one "brain". Using the "standalone" instructions above you could make multiple chips from one Arduino and embed them all in your one project.
If not then how I will make that project run without Arduino?
There are other Arduino boards available that are designed to be embedded in your project. These are much smaller than the "development" boards like the Uno etc. They are also cheaper, being typically the absolute bare minimum you need to get you going. These can be a great way of saving time compared to the Standalone system above, and at the same time saving money compared to using the full blown Arduino boards. Boards include the Mini and Nano range of boards.