Hi Jennifer,
From your question it sounds like the key concept to understand is that in Collector, you are editing (collecting) point features in a feature service. This feature service is contained within a web map, which is the item you selected when you launched the Collector app on your smartphone or tablet. In order to get the same points you collected into a new Collector project, you just need to add the feature service you have edited thus far into a new web map, where it can be combined with other layers, etc.
To do this, start by logging into ArcGIS Online in a web browser. Go to the group under which you found the web map in Collector. There, you should find a web map with the same name as the one you selected in Collector. Open that web map's properties, where you should find a list of layers. Click on the name of the layer that you edited with Collector the first time around. In your URL address bar, you should see a URL something like:
https://www.[your organization].com/arcgis/rest/services/[folder name]/[editable layer name]/FeatureServer/0
Copy the text of the URL up to and including the word FeatureServer (don't copy anything after that). From there, you can go back to ArcGIS Online, click the "Map" tab on the top, and you will have a new web map. From the "Add" drop-down, select "Add Layer from the Web" and paste the URL you copied earlier. That should add the editable layer to your new web map. From there, you can configure this new web map however you like, save it, and share it with your group. You should see this map when you launch Collector (may need to refresh) and it will have all the data you collected previously in it!
Hope this helps. Happy collecting.
Micah
