Hello, I have a compacq desktop that went belly up motherboard wise and the hard drive has a bunch of family pictures that I would like to recover. I am not an expert but could this be explained in laymens terms. I have another computer to use.

There is multiple ways to do this but the easiest is going to be this an external enclosure.
Newegg.com - MASSCOOL UHB-UPS255 Aluminum 2.5" Black USB 2.0 External Enclosure - External Enclosures

Will cost 18$ + shipping etc.

You will have to remove the hard drive from your laptop to put into this device, then it hooks to a PC via USB cable and you will have full access.

Im suggesting this one because you have not listed whether your hard drive is IDE or SATA. So this one does both, and is still very reasonably priced.

With another computer to use, as mentioned above, an enclosure would be an option to hook old drive to working PC

If familiar enough, you (or someone you trust) could put the old drive in PC that is working as a slave drive and have access to that information