I cannot stress enough how important it is to have a mobile compatible version of a website. I wrote several times before on responsive design, sharing some tips and tricks. Here at Blueprint, I do a lot of responsive web development. While I work on a project, I find myself needing to test my work. Here are some tools that are useful in testing websites in mobile.
iOS Simulator
This app is available if you install Xcode, an app available from Apple App Store for free that lets developers create mobile applications. Once you run the iOS Simulator, you will be able to use Safari to simulate browsing on a mobile device. The application simulates both the iPhone and iPad. It is truly useful in testing responsive web designs because you could test how the website looks in both mobile phone and tablet environments.
Android Emulator
This is available if you download Android SDK (Software Development Kit), which is available from Google for free on the Android Developers site. This tool lets you run an Android device on your computer. The Android Emulator also lets you access the web browser on an Android phone.
If you have the real thing (e.g. iPhone, iPad, HTC One, etc.), you should go ahead and use those to test responsive websites. I have an iPhone 5, and I use it for testing websites that I work on. Although the tools above are useful in testing responsive websites, they are not as robust as a real mobile device.
Whatever the case, it is important to make sure to test websites in the development process. Therefore these tools, the iOS Simulator and the Android Emulator will come in very handy in testing responsive designs.