Not sure how to track referalls from partner website

[Mr3putt]

Our company's site is mostly a brochure site for our services but part of the business is signing people up for class registrations.

We have about 30 different events and sign up around 4-8 people a day for various classes.

We have partnered with another website (for clarity I will call this website theirdomain.com) and I'd like to setup tracking so I can see what people signed up for a class on our site coming from theirdomain.com.

I have a Goal setup in Google Analytics called "Course Sign Ups" that tracks all the completions of people signing up for a class. The Destination goal URL is /thank-you/\?(.*) a dynamically generated thank you page with a confirmation. This works fine and is tracking correctly but in Google Analytics how could I figure out exactly who signed up for a class coming from theirdomain.com? Obviously I could look on a day to day basis and see the number of goal completions that came from theirdomain.com but if there were 4 other sign ups that day I have no way of knowing which sign up came from theirdomain.com.

[RhinoFish]

In Google Analytics, left nav, Acquisition, All Referrals.
Above the graph, Goal Set 1.

You'll see all of the referring sources, by domain.
Click on theirdomain.com, so you're just looking at it by itself.

Just above graph, "select a metric", add the specific goal of interest.

Careful, Analytics uses LastIn attribution by default.
If two days ago, they came from theirdomain.com...
Then yesterday they googled your name, and clicked the organic listing, that second visit will get 100% of the conversion credit, it was the last click before the goal action.
Drill into MCF (Conversions / Multi-Channel Funnels / Assisted Conversions, then set the Primary Dimension to Source) for deeper insights into this particular referring traffic, besides being the LastIn click, it might also be credited there with Assists Clicks (all non-Last clicks in the trail that lead to a Goal completion).

And realize that previous clicks outside the time window (defaults to 30 days) or untracked clicks (people switch devices) will likely contribute to referring, introductory, valuable, early-funnel traffic being under counted completely.

If this partner is paid, or if they promote you via email, strongly suggest you encourage them to tag the outbound URLs that point to you, so they properly record their contribution. Better yet, send them the tagged URL they should use.
Google Analytics URL Builder:
[support.google.com...]