I have always done so on foam core boards or on a 4'x8' white board that I have.
I've recently tried using a moodboard app for the iPad, but I still getting used to it. I do use Evernote with shared notebooks to communicate with a team, and make pdf files I can drop into it. Recently started using Penultimate and Skitch to compliment that. If you haven't heard of Evernote google it - it's a free app that runs on your desktop and mobile devices and keeps everything synced up. I did get the premium version, which costs a little, but not much.
But, for me, at some point it always comes down to the physical:
I have a folder on my computer with thousands of inspirational images. I usually comb through it and pull relevant images, then arrange them in bridge. But since making a PDF in bridge is sort of rudimentary, I usually build them page-by-page in photoshop with images and explanatory text, then merge those files into a PDF in bridge. Complete pain in the ass, haha!
I've heard several people mention Pinterest, and I still have no idea what it is, but that word popped up in my dreams last night so I'm going to take it as a sign and look into it!