"Away" or AFK just means you've stepped away from your keyboard for a moment. It's polite to activate this if anyone is nearby, because it lets them know why you aren't responding to their remarks to you.
When Away mode is activated, your avatar appears to "fall asleep", and the word (Away) appears in the nametag over your head. Some viewers give you the option to also force your avatar to sit while Away, which may be a bit more realistic than the "standing slump". If your Animation Overrider contains high priority animations, it will override the Away "napping" animation, making it harder for people to see that you are Away.
|Stop Poking Me! Can't You See I'm Sleeping?|
|Busy Mode Notification Options|
All in all, I very seldom use Busy mode. If someone calls and I'm busy, I just type a brusque "sry, busy ATM" and get on with it. Sometimes, doing things yourself is easier, quicker, and safer than using an automated feature. If I simply must have some solitude, I can log in with an alt or log in to another, quieter grid, or even use OpenSim in standalone mode to log into a region completely disconnected from the internet.