The options is to export the images from Lightroom and then move to the other computer.
However if you don't want to export the images to your computer, you may use the Dropbox Plugin to export the images to Dropbox and then access it from another computer.
You can't transfer edited photos to another computer without exporting. The problem with exporting needs to be fixed. Can you describe the problem in more detail?
My computer tells me there isn't enought memory to export the file that I have worked on. The problem is that the IT guy replaced my OS with a 32 bit version. The computer came to me as a 64 bit.
So I haven't been able to export my work.
As it is a space issue, then you may export the images from Lightroom to an External Drive, and then take it to the other computer.
Please have a look at this article and you may setup External drive in the Publish Services: https://helpx.adobe.com/lightroom/help/export-hard-drive-using-publish.html
It is not a DISK SPACE issue.
It is a memory issue. You do not have enough MEMORY.
Temporarily, you should try exporting with all other applications closed. In the long run, you need to have BOTH a 64bit operating system and 8 GB of memory.
If I were you, I'd ask your IT guy to change the computer back to a 64 bit operating system.