I need to create a registration form for people to chose one of three events, and leave their name, so that when I download the results, I can have a list of registered people for each event.
You could explore one of SurveyGizmo’s pre-built templates for this. When you click to ‘create new survey’ you can select this option:
However, ultimately what you are trying to do could likely be accomplished with one Radion Button Question where the respondent selects their event of choice, and another Contact From question where there respondent provides their contact info.
I hope this gives you some options!