I have a new laptop (running Lion) and want to transfer some of my files from my old laptop (running Snow Leopard). How can I use the airdrop function to copy the files from Snow Leopard to Lion?
AirDrop only works between computers running Lion. You'll need to use another method to transfer files between Snow Leopard and Lion.
File sharing on your local network is one easy way, copying to a USB key or other external storage device is another.
As mentioned above, AirDrop is Lion only.
Depending upon how much you wish to move to your new hardware, the Migration Assistant (in Applications/Utilities) may be of use. If you only wish to move a few files, use either external storage as a "sneaker net" or direct networking as suggested above.