Aseprite
This seems to be the most popular editor available right now. Packed full of features, actively developed, and available for Windows, Mac, and Linux. On top of that, it's open source and can be used for free if compiled from the source code. If you're serious about making pixel art and don't already have an editor you're attached to, this is probably the way to go.
Honestly, if you can get hold of it I can't recommend Aseprite enough as the pixel art creator of choice. It keeps getting better with every update too.
What do I do?
Choose a palette from Lospec
This is a great of making your game look professional and taking advantage of someone else doing the colour theory for you!
What do I do? Watch the tutorial and familiarise yourself with Aseprite.
What do I do? Watch the tutorial and learn how to use the tools in Aseprite.
Quick video showing you how layers and frames work.
Try to think of them like this:
Frames: Are periods of time, moving forward and backwards through frames is looking through time
Layers: Imagine looking top down, the top layer is above everything else and each layer works further down.
What do I do?
Watch the video and learn how to create a basic idle for your character
What do I do?
Watch the video and learn how to create a basic 4 frame walk for your character.
Remember! Walk cycles are kind of a right of passage for animators, they seem daunting but the more you do them the better you will get.
What do I do?
Watch the video and learn how to export your files. Let's show off your hard work!