Without context, it's difficult to comment.

For "extended casing," context is absolutely necessary, for instance. Without context, the term is meaningless.

"Casing," or "case," or "brass" all mean the same thing: The hollow rear portion of a cartridge, which holds the bullet, the propellant, and the primer.
The "bullet" is the projectile which exits the firearm and puts a hole into something.
The "propellant," also called the "powder" or the "gunpowder," is a fast-burning mixture or compound which generates gaseous matter when ignited.
The "primer" ignites the propellant when it is struck by the firearm's firing pin or "striker."
The "cartridge" is the self-contained assemblage of case, primer, propellant, and bullet. This is the "live round" to which you refer.

Each different size of cartridge is known by one or more different names. The name usually includes information about the bullet's diameter, and also some indication detailing which firearm, or firearms, the particular cartridge fits.
Examples: .38 Smith & Wesson; .38 Special; .38 Colt New Service (All of these are different, and not interchangeable.) The ".38" part indicates the bullet's approximate diameter, and the name part which follows the number explains the length and diameter of each different case.

A Colt's-made pistol can be found which fires the .38 Smith & Wesson cartridge, even though Colt's and Smith & Wesson are separate, competing gun manufacturers. Smith & Wesson designed the .38 Special cartridge, but Colt's (and almost every other pistol manufacturer) makes guns which require this particular cartridge.
The cartridge is not considered to be a part of the gun. It is merely something which is put into a gun, and not an intrinsic part of it. Therefore, it is not strange that a .38 Smith & Wesson cartridge would be put into a Colt's pistol that had been properly manufactured to accept it.

"Speer" is a manufacturer of ammunition. They produce bullets, as well as complete cartridges. A complete and unfired cartridge marked "Speer" could be of any size, to fit either pistol or rifle.
Usually, the cartridge case is marked on its base to indicate the "caliber" (that is, its size, and what chamber it fits). The only exceptions are the various lengths of the tiny .22 (or even .17) rimfire cartridge.

For "cartridge," click on: Cartridge (firearms) - Wikipedia, the free encyclopedia
For "rimfire cartridge," click on: Rimfire ammunition - Wikipedia, the free encyclopedia

The exact wording is that a Colt Cobra .38 was found loaded with 1 expended casing marked Smith & Wesson, and other live rounds marked "Speer".

I have now an understanding that an expended casing is the outer shell of the bullet left inside the gun after firing it. The live rounds are those bullets that are still intact and not fired yet. See, without proper knowledge its difficult to know what I was reading, cause so many different names are used, like round, bullet, cartridge and then I had never heard about casing. Thanks so much for your time Steve I took a look at your post and the links you shared. I got a good idea of it.

Originally Posted by MichaelPureSoul

...I have now an understanding that an expended casing is the outer shell of the bullet left inside the gun after firing it. The live rounds are those bullets that are still intact and not fired yet...[emphasis added]

You are using the word "bullet" incorrectly.
The "bullet" is only the projectile. That is, it's the part that leaves the gun to make a hole in something.
Where you have used "bullet," in the quote above, instead substitute the word "cartridge" and you'll have the terminology correct.

Originally Posted by MichaelPureSoul

The exact wording is that a Colt Cobra .38 was found loaded with 1 expended casing marked Smith & Wesson, and other live rounds marked "Speer".

The gun manufacturer/company name is Colt. The model is the Cobra, which is a 6-shot, aluminum-alloy-frame revolver. The caliber is .38 (shortened from the more complete name of ".38 Special").

On a revolver of this type, the revolving cylinder is swung open on a pivot to load, unload, or check and see if it is loaded. It is a 6-shot revolver, so there are 6 holes in the cylinder (called chambers), one for each cartridge (or "round" of ammunition). If you pulled on the cylinder release lever, and then swung the cylinder open to check and see how it is loaded, you might see something like this:



The brass-colored cartridge casing was made by the Smith & Wesson company (S&W, who have made both guns AND ammunition in past years), and it has been fired (or "expended"), as indicated by the dimple in the center of the primer, located in the middle of the cartridge's base. The other 5 nickel-plated cartridges are labeled "Speer" (by the ammunition company of that name, now called CCI/Speer), and they have not been fired/expended, as indicated by the smooth, undimpled primers in the center of each cartridge's base. The color of the cartridge casings doesn't really mean anything in this example; I'm relatively sure that both companies have made cartridges in both plain brass and nickel-plated brass casings at some time during their production. These were just the examples I could find.

In this case, the photo is partially real and partially fake, as I had a .38 revolver in my safe, and a fired .38 caliber S&W-brand case in my fired brass box, but no Speer casings. So I searched online for a photo of a Speer-brand .38 case, and to my surprise, actually found one. Then I made 5 copies of it in a photo manipulation program, and transferred each of the copies over the empty chambers in a photo of my own .38 revolver (it's actually a S&W revolver, not a Colt, but that's difficult to know from the angle of the photo unless you're pretty knowledgeable about revolvers). The result is a pretty close copy of what the written description, above, is trying to illustrate with words.

Thank you so much for that picture!

Originally Posted by MichaelPureSoul

Thank you so much for that picture!

The terminology can be mind boggling. Look up some of the American cartridge types and terms compaired to European types and sizes. I don't much like the Metric system, but when it come to ammunition it makes things easier by giving you an idea how large or small they are. Take a standard American cartridge, the .30-06. It is a .30 caliber bullet, between a quarter and three-eighths of an inch in diameter, that came out in the year of 1906. Fine, but it really does not give me an idea of what size it is. In Metric it would be the 7.62mm (bullet diameter) x 63mm (case length). Now, I can wrap my mind around that and formulate a picture as to how long and such this complete cartridge (bullet and casing together) actually is. This is one way how the media will fool people into believing things about firearms and such that are just not true.
Take the 7.62x39mm round. This is what the "scary" AK-47 shoots. Now, the media will make you believe that this is some kind of super-duper high powered cartridge that will slaughter Hippos and Cape Buffalo as well as people, but this is not the case. The Russian round is called the M43, for the year it came out. It is a medium powered cartridge that is not known for superior accuracy. True, it was designed to kill human sized targets, but it is not the all-powerful round that the news tries to "spin" it as.
Some of the best and trustworthy people out there are gun owners. When I say gun owners I mean the ones of us who actually work for the money to buy the weapons we have-legally. Hollywood and the media try to make the gun out to be something that it is not. Like in the movies where some dude whips out his piece and all fall before him like the thing is some kind of magic wand. BS. It is just a tool that can be used for good and evil, like an automobile for instance. I could probably take out more people with my beefed up Jeep than I could with any firearm I have-not to mention get away fast too! I am glad to see someone, such as yourself, actually take the time to ask some of us who know and use firearms, what is myth and what is reality. Now, if we could just get the rest of society on board, it would make our and their lives easier. JMHO.

Sorry about the above post, I used indentations for paragraphs, but for some reason it did not come out!

Thanks ^^ great points!! The media spins everything, it's hard to determine what is really true and what not these days.

I made the right decision to come here and ask, who else but the right people