The specs for my frame state only: B.Bracket: EN BB68
I understand that is the standard English BB.
Will this bottom bracket fit: BB 68 X 115.5mm?
What does the 115.5mm stand for?

That's the length of the BB spindle. You need this to correctly match your crankset for proper chainline.

So, choosing a BB is two parts. The threading (and width) of the BB cartridge needs to match that of your frame's shell. Then the spindle length and type of interface must be compatible with your crankset.

If you're replacing a worn BB, measure it (or look for the markings) to determine the correct spindle length.

The 115.5mm is the spindle length of the bottom bracket. It will fit in your frame but may not work with the crankset you're planning to use. Different cranksets recommend bb spindle lengths specific to their installation.

It is for a new frame, and I am looking at this Ebay buy: https://cgi.ebay.com/ws/eBayISAPI.dll...:B:WNA:US:1123
So what you are saying means the square tapers of this BB (spindle?) have a width of 115.5mm.
Is that a good width for a Compact Campy crank of some sort?

Another question: BB30 is out, when I am getting a En BB68 frame, right? I have to stick with English 68mm.

This one https://www.ebay.com/itm/SRAM-TRUVATI...95990176507427
on the other hand has nothing sticking out like the square taper of the above Veloce!

Approaching it the other way round and buying the crank set first, what BB do I need then?
This is what I am looking at: https://www.ebay.com/itm/Campagnolo-R...49305255951469

The crankset in your last post has an integrated spindle which is designed to work with external BB cups (e.g., Campagnolo Ultra Torque BB cups).

To use BB30 bottom brackets and cranks, one needs a frame built with a BB30 bottom bracket shell. It will not work with a frame with English-threaded 68mm BB shell.

Those eBay items you linked to are actually three very different Bottom Bracket / Crank systems, and their components are not interchangeable / compatible with each other.

The Campy Veloce bottom bracket you linked to is an internal bottom bracket and intended for use with three-piece cranks. The bearings are inside the the BB's central housing that sits inside the bike frame's BB shell.

The SRAM GXP bottom bracket you linked to is an external-bearing bottom bracket, where the bearings are outside of the bike frame's BB shell. It will only work with SRAM/Truvative/Bontrager 2-piece stepped-spindle road cranksets.

The Campy Record 11 crankset you linked to is also an external-bearing 2-piece system I believe. It will only work with Campy external bottom bracket cups like these: https://www.amazon.com/gp/product/B00...M0EYG255WH89BV

So remember you are not only buying a bottom bracket-- You are buying into a particular BB / Crank system.

Thanks for all that info! I can see that finding all the parts to work together on Ebay may be quite a difficult process. But buying a new group set is just so expensive that I cringe looking at them.