A J tube is exactly that: a tube shaped like a J, extending from the top of a jacket to the mudline (or close to the mudline), with a bell mouth on the end, to provide a conduit to pull risers up.  The radius of the J is a function of whether the risers to be pulled up the J tube are flexible or (more normally) rigid pipe- on the platform I'm on at the moment, the 3 J tubes' radius is about 100ft, and they are each 16" OD.

To expand on the description. In submarine pipelaying, the normal method in shallower water is to S lay the pipe, where the pipe is welded horizontally (or near horizontally) on a lay barge and runs off a stinger in an S (reverse) curve.

In deeper water the J lay method can be used where the pipe is welded in the vertical and there is a single curve near the seabed.

On the platforms that I work with, a J tube houses an umbilical or flexible flowline from the bottom part of our hull up to the topsides.  In my case the platform is a Tension Leg Platform in deep water.  For installation, the umbilical or flowline is connected to a "pull head" that is used to pull the umbilical through the J tube up to the topsides.  The J tube is curved at the bottom so that it follows the curve of the umbilical as it leaves the platform.

The OD of the tube and the bend radius need to be large enough to allow the pull head to be pulled all the way through the tube.  The wall thickness should be such that the pipe can resist wave/current loads and any loads applied by the umbilical or flowline.  The length of the tube in my case is around 170'. Typical sizes in my application are 16" to 22" OD and approximately 3/4" thick. You may also need flanges on the tube and supports to hold it to the platform.