At a Pro/Am comp a Scholarship event is one where the student actually wins money to be used for future lessons. At some competitions the check is written out in the student's name, in some it's to the teacher.
Different comps have different rules for scholarship event elegibility. For some, you just pay the entry fee and show up. For others, you are also required to enter a minimum number of single-dance (also known as "group", also known as "freestyle") events to "qualify." You don't have to have any particular placings, you just have to enter. At some comps the minimum is as low as 4 dances, the most I've ever seen was 8. You can usually check the rules for the scholarships on the competition's web site. The rules will also be in the information packet that gets mailed out to various studios and professionals.
Many comps offer separate Bronze and Silver scholarships in each style, plus a "Open" scholarship for the more advanced Pro/Am students. Some comps combine the Bronze and Silver into a single "Closed" scholarship. Sometimes Gold level is lumped into the "Closed," but usually not. Occasionally there is a separate "Gold" level scholarship. Again you'd have to check the rules for the particular competition you are entering.
The money isn't really all that great. In fact, it barely covers the cost of entering the required single-dance events once you take into account your teacher's per-dance fees, ballroom tickets, and so on. Still, it's better than a kick in the head

The money ranges from $50 to as much as $500, depending on the competition and your final placing.
The nice thing about dancing in the scholarship events is that they are much more competitive than the single-dance events. Because the single-dance events are divided up by gender, age, and syllabus sub-level (the interminable "Pre-Bronze - Bronze I - Bronze II - Open Bronze - Silver I" kind of series you'll often see), you'll often end up dancing against very few people. There were so many times when I was dancing Pro/Am that I considered myself lucky if there there three couples actually competing against me....However, in the Scholarship, all those sub-divisions are collapsed. Male students and female students compete against each other in the same event. All age groups compete against each other. All the level sub-divisions are collapsed into one. There have been many times that there have been so many people in my scholarship even that there will be a semi-final or even a quarter-final. Now *that's* competition! It's very exciting, and if you make a cut or two you end up getting a lot of floor time for your money.