There IS a name for it, but I forget right off. It acts as a kind of trim board, because the last thing you want is for the ends of the boards to butt up to the tile.

That is commonly called a header board Lisa. Header boards should be joined with the field using a tongue and groove method. A gap should be left where it meets the tile and that gap later gets filled with a color matched grout caulk. Like below....

Here is a link that might be useful: wood to tile transitions

Nice website, Jerry. What area are you in?

And how do you make some of those radius cuts so perfectly? Amazing work.

Thank you both so much, this is very helpful. Is it a professional task, or can you DIY?

Anyone can do it Lisa. If you need a header transition on the right side you would square it and install the header with the groove facing out. If the header is on your left, install it with the tongue facing out. When you get to the headers start your rows off the headers, working right to left or vice-versa. Sometimes there will two header transitions opposite from each other and you would need a router.

I'm in SW Virginia homebound. I use a piece of MDF to create a template and place the template down on the flooring and then cut the radius using a router.

Ah yes, of course. Still, not for the faint of heart! Nicely done.

So presumably a shallow router bit is used for inlays, too?

Thank you homebound. Its best to use a plunge router with a 1/2" shank and cut down in increments. The first pass should be around an eighth of an inch deep or no more than 1/3 of the thickness to keep from hitting any hidden fasteners.