Pro/Am is where a student (the amateur) competes with his or her teacher (the professional). It's quite popular in the US as a way for women to compete who otherwise can't find partners. They'll take private lessons with their teacher (so it's just the student and teacher working together) and then when the student is ready they'll enter a Pro/Am competition. The Pro/Am couples compete against each other, not against Pro or regular Amateur couples.
In the Pro/Am syllabus divisions, only the student is judged, but in the Open-level division, the professional and amateur are judged together as a couple.
There are two forms of Pro/Am events: single-dance (or "freestyle") events and multi-dance events (which are usually scholarship events awarding a bit of money to the winning couples so the student can continue his or her training). In the single-dance events, male students are judged separately from female students, and furthermore the events are divided up by age and syllabus level. So, there's a place for anyone and everyone to participate. The multi-dance events are usually only divided by syllabus level, and male and female students compete against each other.
I competed in Pro/Am for about seven years, while also dancing in various regular Amateur partnerships. I don't do Pro/Am any more but still compete as an Amateur. So, if anyone has other questions about how things work or how they compare, ask and I'll be glad to answer!