A connection is established when you connect an app to your organization. To connect an app (or API or other online service), you simply authorize Better Reports to access your app's data. Once the connection is established, Better Reports will start importing your app's data. Depending on the amount of data, the initial import might take anywhere from a few seconds to several hours. However, the initial import is a one-off operation. Once fully imported, Better Reports is constantly working in the background to keep your data in sync and reflect any data changes that might occur in the connected app.
Connections are what you get billed for each month and each connector has its own pricing model.