Before we get into some examples, here's my take on Design thinking: at its core, you're designing a solution that'll work better for Humans. 

If something you made doesn't work for Humans (you, me and other interesting folk evolved from apes), it's not going to succeed. And this applies to every field - from technology, to food, to healthcare, to human resources - any and every place innovation is possible.

The Design Thinking Methodology

BASIC PRINCIPLE: You'll never, ever, ever, jump to a solution early. If a million light bulbs light over your head, an exclamation mark appears in thin air over your head and you feel the urge to yell 'eureka' and run, erm, down the street. Don't.

RESEARCH: You'll observe, really observe, how people behave and interact in the area you're researching. You'll try and be inconspicuous, and learn. For example, say you want to create the world's best Champagne glass. Instead of heading to your industrial designers and dictating "Make me a glass!” design-thinking-you will head to places where you can observe people interacting with champagne. Could be one of the swanky post-Oscars galas (!) filled with the world's best entertainers. Or could be the dorm at Northwestern University the night after graduation filled with celebrating undergrads. You'll then observe. How they grip the champagne glass. Where they put it. How often they drop it. Do they fill it to the top? Do they even use a glass? How long does a glass stay in a person's hands? Etc. etc.

DESIGN: Good observations will bring you to good insights. You'll then use the insights to design - rapidly. Say you observed that most people grip the glass by its rim, rather than the stem. Your insight? You need to do something tremendously awesome with the rim of the glass that makes it look better and slip less when held from the rim. Now is when you design your product.  And they key here is rapid prototyping. You build a model, take it to a quick focus group. People think your Champagne glass 2.0 is too top-heavy. Back to the drawing board. Back to research. Back to observations. It's a blurred mishmash or high-speed, human centered design.

BUILD: Finally, you've got the best design possible that is amazing in every sense - usability, looks and awesomeness. 

And that's Design Thinking in a nutshell. Phew. Not rocket science, is it? (for a more professional explanation - try this)