What are transcriptions?

In musical terms, a transcription is a piece of written music that is based off of a recording.  The transcriber makes his best effort to accurately write down exactly what is being played on the recording.  Most often people will transcribe improvised solos from Jazz music to improve their own improvising skills.  Others, like yours truly, transcribe whole songs that aren't available elsewhere as sheet music.

What equipment do you use?

Usually I use my iPod to listen to the recording at the piano, headphones so I can hear individual notes better, Finale to notate the songs on the computer, and Acrobat to port the Finale files to PDF format for cross-platform compatibility.  A handy tool for those of you just starting out is the Amazing Slow Downer which slows down audio without changing the pitch.  I used that for my first song but I don't need it anymore.  If you choose to use that software, I advise that you don't get dependant upon it.  It's not good practice. :)

How long does it take to transcribe a song?

That depends on the duration and complexity of the song.  I find that usually I take one hour to transcribe one minute of music.  That's a long time, I know, but I'm obsessed with perfection. :) I don't know how long it takes others to transcribe, but I imagine that my hour-to-minute thing is a reasonable time to take to get things right.