Elements: How do I transfer my application from my Win 8 laptop to my Win 10 desktop?

You own it so I think all you need to do is go to the Adobe website and download another copy to your Win 10. You will need the number, which you may have to get from the maker of the laptop if they didn't send the disk and box.

Elements 8 is a quite old version. I remember purchasing Elements 9 several years ago, I believe in 2010...
 
If you can't find the Elements 8 installation files (you won't find them on the Adobe download page...)  you probably need to purchase a newer version (Adobe is currently at version 16, a.k.a. Elements 2018).
To transfer your catalog with all the thumbnails, tags, albums and metadata you need to copy the complete (with all subdirectories) catalog directory to your new desktop. The default location of this directory is C:\ProgramData\Adobe\Elements Organizer\Catalogs . If you put your catalog directory here, Elements will show this catalog when you navigate to the Catalog Manager.   
You have then the possibility to convert (update) your old catalog to the current version.

I am not sure you can reinstall and activate PSE8 on another computer.
1 - It looks like you got the programme installed as a 'bundle' when you bought your computer. Then, the license may be an oem license linked to your specific hardware. You may not have an install folder left on your computer, if there ever was one.

2 - See:
https://helpx.adobe.com/photoshop-elements/kb/download-earlier-version-photoshop-elements.html

The problem will be that you would need the license key to activate the software.

Anyway, it's a software from 2009 and you may have big problems with the current OS versions and hardware.

An important advice:
If you migrate your catalogs and your photo files, read this to move them safely to a new computer via a backup and a restore:
https://helpx.adobe.com/photoshop-elements/kb/backup-restore-move-catalog-photoshop.html

Edit:
Jack,
We were writing at the same time...

Thanks for your replies.

I downloaded it but I get error messages when I try to extract the Zipped folder (!   C:\Users\Steve\AppData\Local\Temp\PhotoshopElements_8_LS8.7z: Cannot open Data1.cab, etc).

Although I was happy enough to continue using PSE 8 it looks like I will need to invest in a newer version to use it in Windows 10. So I don't mind using an old version of PSE but does anyone know which I can buy that will work with Windows 10?

Just tried another app to extract the zipped folder and it seemed to work except that when I put in the serial number during the installation process it wouldn't accept it.

The serial number should be 24 digits.
You can't recover it from the truncated 20 digits number shown in the opening window.
How did you retrieve the serial number?

From the System Info within the program's Help. But, yes, only 20 digits. I have an Adobe account and I have tried to there to help me find the number and I can only find reference to PSE15. So where do I go from here?

Hi Stephen,  

Windows 10 should upgrade from Windows 7 with the apps still on there, and you may need to run compatiblity for that one.  Get that from right clicking the Adobe program icon then select compatibility.   

Also, if you need the program key to reinstall your Elements 8, and if you registered that at Adobe, you can get the product key through these steps:   

Start by signing in the Adobe site:  https://accounts.adobe.com
Sign in using your Adobe name and password at the link above, 
then go to Manage Account, next click Plan and Products, 
then click View My Products, get your licensed products and keys. 

I think Michel had this but without the link to it, so follow Michel's coordinates for that adobe page.   

Steve Lehman   

Thank you all once again. Tried as suggested but, although I registered the other day, I just get the message "You have no registered products." and of course it rejects my 20 digit number.

It seems your preinstalled Elements 8 came with a limited or restricted kind of installation key. Therefore you cannot install it on another system and Adobe is not aware of your registration.

Maybe you should try to remove the HDD of your old desktop and install in in your new desktop. With a little luck you might get it to work. Alternatively, you can use one of the many HDD copy utilities which allow you to duplicate your complete HDD to an new/other HDD (as long as the data size of the old disk is less or same). 
If you have an Windows 10 installation disk or set, you can then upgrade your Windows 7 OS to Windows 10.

Jack and Stephen Poole,   

Reinstalling an old drive is a long process if you want that, and the old drive may be 32 bit and could have more problems with setting up a dual-boot for each drive.  Also, the ribbon cable inside will need to be used differently than the Windows 7 config as its CDrom drive may currently be on the ribbon cable's slave connector, so where do you connect the old drive?   And then, it may need its own motherboard adaptor for both drives to talk properly, otherwise, they won't "see" each other.  All of this is real complex, and the real problem is about software compatibility with Windows 10.   

Might I suggest Stephen upgrade to a newer Elements as an easier solution.  The newer Elements are like the older Elements with the only difference being that, the back button is at the bottom instead of the top.  It's only an mouse up/down stroke to get used to.   It should have all the same tools as Elements 8, and you might even pick up a few new tools.  I have Elements 8 and 10 which are the same, so I uninstalled version 8.   I have 2, 3, 4, 8, 10, 14, and latest.   Stephen, use newer software.  You won't regret going forward, and you'll have more luck.   At Microsoft, we had an old motto (1999), "keep current" although I keep the old stuff on mine too.   

Steve Lehman, mcse   

I have decided to bite the bullet and upgrade but thank you all for your help and suggestions - much appreciated,
Steve