Not much to say about NG+ really, it's just like Demon's Souls. As soon as the credits are done rolling, they toss you right back into your jail cell with all of your gear. You lose practically all of your key items, like embers, repair boxes, keys, some rings, etc. Estus flask upgrades carry over, so before you finish your first playthrough, you'll probably want to collect every Fire Keeper soul that's available. Enemies hit harder, have more HP, and give more souls, though the percentage increase varies between each enemy. Weaker enemies get the biggest boost, while the stronger ones get a smaller boost. I think everyone gives 3x as many souls as before though.
The guide says that in your first NG+, enemies' health values increase from 1.5 to 2.5x, attack power increases from 1.3 to 2.3x, and defense increases by 1.6x, or remains unchanged. Then in every subsequent NG+ run, all values increase from 1.07 to 1.25x, until it tops out at NG+6.