Android is great about multitasking in the sense that background applications will be suspended where you left them. That means you can go back and look at the previous app you were using, but it's going to be in full-screen covering whatever you're trying to look at now. This makes note-taking for research really difficult when you're always having to tab back and forth and can only see one app at a time.
Hovernote seeks to remedy this issue by adding a floating notepad to your Android tablet screen. That way, you can read a Web site and take your notes at the same time.
Download a copy of Hovernote for your Android device. Despite being optimal for tablets, you can use this app on any Android device -- though I'm not sure how useful it will be on a smaller screen.
Tap on the small h icon in the notification area. On your tablet, this will likely be the bottom right-hand corner. On a phone, it will be at the top in the notification shade.
You can start typing right away, but you can also make a couple of adjustments if you like. For starters, the window can be dragged around the screen by pressing and holding on the hovernote text at the bottom of the window. That way you can move it out of the way if what you're reading is being covered. You can also change the size of the window by pressing and holding on the dots in the bottom right-hand corner. Luckily the window even allows copy and pasting functions, so you can clip quotes from the Web.
When you're done taking notes, you can save them by sharing to Google Drive, Dropbox, Gmail, or several other services. And when you're ready to close the app, just tap the menu button in the bottom left-hand corner and then the X.