It's called welting or cording
ETA: Depending on the size of cord used under fabric it can be larger or smaller.

I believe it's a single cushion Lawson style sofa with a leather slip cover or made to have slip cover look.

This post was edited by chijim on Mon, Nov 3, 14 at 12:53

I'm glad someone else answered. I wanted to say extra big cording, but I don't really know and didn't want to bump your post down.

Thanks. I was unable to come up with even 1 example on Houzz or Pinterest.

Is this out of style?

I have a strong suspicion that that sofa was designed and custom-made. You'll see several sofas in the link below that have corded, or welted, seams, none of them leather. The cording on the sofa in your photo is extra large, I think, because leather is much thicker and more difficult to work with than fabric. You don't usually see (I've never seen) welted seams that large. Unless you're specifically looking for a leather couch, you can probably find a decent selection with that look.

Here is a link that might be useful: retro sofas with welted seams

I have always called the sofa where the back and arms are the same height a "low backed" sofa. It does look like a well worn leather slip cover, probably made in the 1970s.

Finding a similar sofa shouldn't be too hard, but unless you sew leather, finding someone to make a slipcover for it may be more difficult and expensive.

Here is a link that might be useful: low backed sofa

That style is called tuxedo style, back and arms the same height. That is either jumbo welting or there's a technique for sewing the top to the boxing strip where it looks like welting, but it isn't. There is a name for this technique, but I can't remember it now.

clt3 is right--Tuxedo. A Lawson has a lower arm.

I have leather upholstered stools and the welt is conventional size, so even though this is leather, the large size of the welt is an added detail.

It does appear to be at least constructed like a slip cover with a welt added at deck height to define the "cushion" area of the sofa, but that all seems to be one piece.

The welting on that sofa almost looks like flanges. I wonder if you searched for "flange seams" you'd find something attractive.

Here is a link that might be useful: Quick search results

The edge detail is called French Mattress Style Cushion.

{{!gwi}}

Here is a link that might be useful: French Rufted Mattress Style Cushion

Lee industries has a leather slip covered sofa somewhat similar. Item LS1297-03. I just noticed it on their website earlier today!

http://www.leeindustries.com/style_detail/lee/id/LS1297-03/in/LS1297_03_0509

Here is a link that might be useful: lee Industries