I've created my buffer, 10 m around surrounding streets, now I need to remove all trees (points) that are within that buffer but I'm having trouble doing so! I've tried clipping but I guess my input/clip features are wrong.
Try "Select by Location" and set the trees as the Target layer. Set the buffer as the Source layer.
Then, for the spatial selection method, use Are within the source layer feature or Are completely within the source layer feature.
That should select all the points within the buffer. Now, you have three options:
1) In edit mode, open the 'Trees' attribute table, right click one of the highlighted records and delete them.
2) Similar to 1) you can easily delete selected features without the need for entering Edit mode by running the Delete Features tool. Be aware if you have no selected features then this tool will default to deleting everything and there is no undo for this action!
3) Open the 'Trees' attribute table, reverse the selection, then right click the layer in the table of contents Data > Export data as a new feature class or shapefile.
As commented by @MichaelMiles-Stimson, instead of using the Clip tool I think you should examine Erase:
Creates a feature class by overlaying the Input Features with the polygons of the Erase Features. Only those portions of the input features falling outside the erase features outside boundaries are copied to the output feature class.
Erase requires an Advanced level license but if you are using Basic or Standard then there are easy workarounds documented on this site.