Thanks to all of you for your input!
As you may have guessed, I'm dealing with a leaky faucet. I think that I've got the leak solved now.
I bought the cartridge shown in the photo, but I ended up not installing the cartridge. The original cartridge is much sturdier, so I used it. I did take the spring and rubber piece from the new cartridge set, and replaced the old ones that were installed in the faucet. The old ones are what are actually shown in the photo. It was only today that I even noticed the spring and rubber piece in the faucet, as they were pushed down, and not very visible.
I did not replace the O-ring.
Bottom-line: all seems to be okay now.
If I need to repair a similar leak in the future, I'll first look to replace the spring and rubber piece, and possibly the O-ring, just as Terry suggested. I'll keep using the original cartridge as long as it works well.