First time I heard about Raspberry Pi, I have to say that I hated the name and because of this I didn't want to hear about it (silly humm...). I also find out it was too techie for me. In all cases I was wrong, it is just that I didn't have the right knowledge at that time to understand how powerful Raspberry Pi can be.
What is a Raspberry Pi?
Compared to Arduino which are micro-controllers, Raspberry Pis are real computers. Exactly the same as the one that you are using in order to read those lines. Raspberry Pis are using cheap but efficient components in order to make them available at an affordable price starting at 5$.
There are somehow hidden costs due to the additional components you need to buy (keyboard, sd card...) though there are many ways that you can find in order to decrease those costs.
By providing you with a cheap computer, you can easily insert a "brain" within projects you have in order to take advantage of the internet of things. Let's imagine that you would like to have your own server within your house in order to control the temperature, check if the tv is on, if your children are doing their homework, if your cat has been fed that's typically what you can do.
Even if you can choose the operating system of your choice, most of Raspberry Pi users are using GNU/Linux operating systems, this operating system is very powerful as it provides a graphical user interface so as a lot of documentation if you want to control it over the terminal. As most of servers in the world are using GNU/Linux operating systems, this is really easy for a geek to control remotely a Raspberry Pi and having fun with it.
One last thing, compared to laptops, desktops and smartphones, Raspberry Pis have ports that you can use in order to plug electronic components on it so as you would do with an Arduino. It is just that the programming language you will use will not be the one of Arduino but probably Python, another programming language.
Ok but what are the differences with Arduino?
If Raspberry Pis are so cool, why not using Raspberry Pis all the time? Well those are two different technologies. Arduino is here in order to make small digital devices, as a result it costs less and consume less energy, it has a big community of makers in order to provide you with a lot of ideas for your projects. Raspberry Pi is nothing more than a computer. So to say, when you start your project, you need to ask yourself how complex/big this project is going to be. According to those answers, you will know if the Raspberry Pi or the Arduino is the right fit for you.