All of these talks have had a large impact on how I approach problems or understand different aspects of the art of programming. Everyone should be exposed to each and every one of these videos if they're looking to become a more well-rounded developer.
If you prefer reading, I've got you covered with this post listing my book recommendations.
I'll start off by listing talks that I think are relevant no matter what problem space you're working in, and add some more targeted talks to different sections below.
Simple Made Easy - Rich Hickey
Rich Hickey is the creator of Clojure and kind of my idol. This talk is relevant for any language, and has some general tips for keeping your programs simple. It's worth a watch.
Hammock Driven Development - Rich Hickey
Rich is back. This time he elaborates on the process of an individual programmer and in what ways you may be able to do better work.
Making Badass Developers - Kathy Sierra
A bit of a cure to impostor syndrome perhaps. Really great talk outlining a way to learn things faster by breaking concepts into smaller chunks.
F*ck You, Pay Me - Mike Monteiro
Fair warning, language here. This talk is specifically targeting designers and dealing with clients but in my opinion is relevant to almost anyone spanning really any discipline. You should watch this to understand 'value' in general. You'll also get some insight into what running a (smaller) business is like sometimes.
What the heck is the event loop anyway? - Philip Roberts
If you're trying to wrap your head around functional programming, map reduce yourself a sandwich. Lots of well communicated examples for the core concepts of functional programming as they can apply to JS.
What Have Namespaces Done for You Lately? - Liz Rice
This talk shows you how to build a very basic container and in doing so will give you a lot of insight into the building blocks behind containers as Liz builds one almost from scratch.
Even More Talks
There are tons of great lists like this around the internet. Many of these are really big and comprehensive. You'll get even more hits searching the ol google for 'awesome talks'.