airplane mode

Just what does the airplane mode do, my daugheter is fixing to take her first plane trip and I need to know so I can explain it to her.

Prevents the phone from accessing cell towers. I used this when traveling overseas to avoid roaming charges. I think turns off wifi at the same time, which you can then turn back on. For flying, either leave the phone off or use the airplane mode and use the phone for reading a book, playing a game, etc.

it's sleep mode, leaves phone on but save the battery
I use it everyday as the cell towers don't work here in the mountains of Va. and the phone will just search and search at full power until the battery is dead very soon, so yes basically it turns off phone use from towers and turns off wi-fi and blue tooth, but those can be turned on individually if you need them.
I can access the internet from my wireless at home even though the phone does not work,by turning on wi-fi

It's not really sleep. It just disables all wireless communications (as required by airlines). As already mentioned, that includes communication with the network (voice and data), WiFi and Bluetooth.