How can I connect a Mac Mini to a VGA monitor?

It depends on which version of the Mac Mini you have. According to this Apple Support KB article, older Mac Minis can connect with a normal DVI port, which you can buy a DVI-VGA adapter for.

While the generation after that offered Mini DVI and Mini DisplayPort outputs, both of which can be adapted to VGA output.

Then immediatly after that there was a generation that offered HDMI and Mini DisplayPort output options, only the Mini DisplayPort could be adapted to VGA output.

The current Mac Mini uses HDMI or Thunderbolt(which is compatible with Mini DisplayPort) for output. Only the Mini DisplayPort / Thunderbolt output can be adpated to VGA output

Why no VGA on HDMI that is found here KB article:

Can the HDMI port drive analog displays (VGA displays for example) using adapters?

No. The HDMI port will drive digital connections—such as DVI—using the adapter that came with the Mac mini, but cannot be used for analog displays such as VGA. You can purchase a Mini DisplayPort to VGA adapter separately to drive analog displays.

For Mac Minis with the Thunderbolt or Mini DisplayPort, the Mini DisplayPort to VGA adapter would be the same as that used by MacBooks.

Third party adapters are also be available and at a lower price usually. A good low price source for compatible cables and adapters is MonoPrice.com

There is a solution to have both computers hooked up to the one display and that is getting a KVM . In your described situation a 2 port USB-VGA-KVM would allow you to share your keyboard, mouse and display between your Mac Mini and your PC.

Additionally you would need to obtain the necessary VGA adapters for your Mac Mini as mentioned in JWO1's answer as well.