This should be pretty basic, but I can't find this information anywhere, so I will ask here:
I want to track the conversion rate for different campaigns that we will be using to advertise our Shopify app.
How do we pass some sort of a tracking parameter to separate our advertising campaigns and measure their efficiency (so not just all conversions, but per campaign)?
You need to add tracking ID information to your app listing. Google Analytics can provide this information once it's set up.
For future reference, this question would be better suited for the Shopify Partners area of the forum.