How to increase the RAM memory of my iMAC 21.5 inch, mid 2010?

Video of the procedure is here:

https://eshop.macsales.com/installvideos/imac_mid_2010_mem/

When we upgraded the RAM in my wife's 2011 21.5-inch iMac, I had my grandson do it. He was about 12 years old at the time and had never taken apart a computer. It took him less than ten minutes and the operation was a success--no issues then or now.

"Just any RAM" won't work right in a modern Mac. My two trusted vendors are:

https://eshop.macsales.com/shop/memory/iMac/2010/DDR3_21.5_27

http://www.crucial.com/

Note in the OWC listing that the entry-level i3 processor used in some 2010 iMacs will not support individual RAM modules greater that 4GB. You can find your processor type from "About this Mac":

Please do take a look at this note re. the 2010 models (part of the document on how to install RAM supplied above) - it depends on your exact model/processor:

i5 and i7 Quad Core iMac computers come with both top memory slots populated. These computers will not start up if only a single DIMM is installed in any bottom slot; these computers should operate normally with a single DIMM installed in any top slot. Core Duo iMac computers should operate normally with a single DIMM installed in any slot, top or bottom. ("Top" and "bottom" slots refer to the orientation of the slots in the pictures below. "Top" refers to the slots closest to the display; "bottom" refers to the slots closest to the stand.)

So, you can install two modules in the bottom row for any of the models listed - there may just be a problem (depending on the processor) only installing one in a bottom slot.

Install memory in an iMac - Apple Support