Once upon a time one could only view a website using a computer screen, and the few mobile phones that existed were used for making voice calls. Then gradually, mobile phones were able to display websites and web designers slowly adapted, sometimes creating a second website version for mobile.
The growth of the internet has resulted in many more devices displaying web pages and designers are not able to create website versions for all these devices. There would be a website for dumb phones, smart phones, tablets, small computers, big screen computers, TVs, etc. The solution is one website that automagically fits to display well on any device.
To get see a live example of this, open this website on your phone, ipad and computer; or resize your computer browser window between the small size and largest size to see how this web page resizes itself to suite the display size. This is referred to as Responsive Web Design, first introduced by Ethan Marcotte on his article "Responsive Web Design".
Ok this is cool but should I adapt it in my website?
A good answer will come from your website statistics where you can see what sort of devices your website visitors are using. (For those who don't like any hint of techie language, jump to the next paragraph). If more than half of your visitors are on a resolution of 1024px by 768px or there abouts, then making your website responsive may not be priority.
If many of your website visitors are using a whole range of devices, then you should list 'responsive web design' on the things to do on your next website revamp.
Will a responsive web design increase my profits?
The benefits may not be that radical but here's a simplistic answer - your website feels and reads better, that means your website visitors will spend more time on your website, getting your information thereby increasing the conversion rate from a casual browser to a potential customer. An old looking website that doesn't display well on various media is a struggle to read through.
For an in-depth read on responsive web design, click here.