They are two different things. Monitor calibration is a profile that is changed by the Settings->Display->Color. It makes sure you see greys as neutrally grey as possible on the monitor.

In PS, you work in color spaces. sRGB has the least amount of color data. Adobe RGB has more data, and ProPhoto has the most. When working with master files, you want to work in the largest space possible, then reduce as needed for final output. So what I do is work in ProPhoto 16bit for master files. Then when I want to print, I convert to Adobe RGB bc printers can print that much data (but not all of the data of ProPhoto color space). But I still edit in ProPhoto for future proofing. It's a bit overkill. Most everything is fine in Adobe 8bit for what you can output in today's technology. And then for Internet websites, I would convert to sRGB color space and then output to JPG.