A furled line in tenkara is a fixed length of material braided together from strands of either monofilament or even cotton thread to form a line. Each furled line starts thick at one end and tapers to a thinner diameter at the end closest to the hook.
A furled line is typically the length of the tenkara rod or even just a bit longer than the overall rod length. There are a few different brands available in particular line available from Tenkara USA, Moonlit Fly Fishing, Streamside Leaders, and Cutthroat Leaders. Each brand has their own distintive features that each brand and style offers.
I suggest that each beginning tenkara angler starts out with a furled line because of the very soft presentation that is an inherent part of these lines. Also a furled line has a bit of weight which helps a new angler learn how to cast a tenkara rod properly before moving on to other lines such as level lines.