How to add mutliple items to a list using the same button?

Question

In general the program I am trying to create is for horse racing.
Okay so I am trying to figure out how to add multiple names to a list from a textbox on button click. I want to be able to use the same textbox and button to continue to add to the list without having to close the window and reopen it. I will need to be able to access the list later for the user to input times and scores for each rider in the list. I'm not so much looking for exactly how to do this but a push in the right direction would be nice. I'm kind of new to programing but I have a pretty firm grasp on C# basics and I want to learn more. Hope someone is able to help me out here a little bit. :)