Being a highly visual person, I often sketch out an idea that I am developing on my own or when collaborating on a project. Since I almost always have my tablet with me, it only seemed logical to use it to sketch in addition to my trusty Moleskine notebooks. Using the tablet to sketch ideas also makes sharing with collaborators much easier, since there is no scanning of paper involved!
iPad
Paper: This is my favorite app, hands down, for sketching on my iPad. I love the way that the app organizes sketches into journals/notebooks and think that the drawing tools have a great natural feel to them. Did I mention that you can also share entire notebooks for great project collaboration? The app is free and comes with a black pen tool; additional drawing tools are available for in-app purchase. Free
Scribable: Great app that's perfect for those ideas that arise during meetings. I discovered this app from a reader recommendation and have to agree that I also find the "continuous notes" concept quite handy. $1.99
ShowMe: If collaboration is key and you'd love an easy way to explain what it is you're thinking while drawing, use this app. This app also has a flourishing community filled with educators using it to explain everything from grammar to algebra. Free
Android
Adobe Ideas: While I mainly use this app for quick visualizations, you can use it for vector drawing as well. It's simple and easy to use and the app can really be as basic or complex as you need it to be. Also available for iPad. $9.99
Whiteboard: For a free whiteboard app on Android, look no further. While it does lack the community that ShowMe has, it is simple to use and built for easy sharing. Free
What apps do you use to sketch an idea?
(Lead Image: Flickr member Jorge Quinteros licensed for use under Creative Commons, all others as credited above)

Shaw's Original Fir...
Well sketching is really a different things than taking notes. Different apps do them both well. I have Paper, and it looks beautiful and organizes things in an interesting way. But sadly, its is almost useless. The biggest positive for the app... they do watercolors better than any other app. Major cons... you can't zoom in, you can't change the size of the brush, there are only a few colors to choose from.
From paper I tried sketchbook pro, but the results were very digital. The charcoal and paint didn't look like the real thing, not even close. I eventually found Art Rage which is far an away my favorite. The materials are amazing, especially the pencil/charcoal and the oil paint. Especially the oil paint. You can save several paintings in progress, export them, drop the files into drop box. Best of all you can zoom in, change brush size and qualities and pin reference images to look at while you are working. Even to sample colors from. Amazing app.
For note taking.. Noteshelf is the best I've found. It organizes your notes into books on a shelf, similar to paper. You can specify what type of paper or format you would like to take notes in. You can share the books, export them out as PDFs or add them to Evernote. You can type with the keyboard, sketch with a stylus or your finger, Highlight, link, take photos and embed them. Excellent functionality.