The Next Appointment column will show the next non-recurring appointment for the lead. If you have a recurring appointment on the lead the icon will be displayed. You can click on the icon to display more details on the leads upcoming schedule. You can also click the My Scheduled Leads button in the header. This will apply a filter that shows you the leads that are assigned to you and have a next appointment. It will also sort the appointments into ascending order so you can view your upcoming appointments in order. By default the Next Appointment column will display on the lead list. If you wanted to not show it you can use the column options and un-check the Next Appointment option or by disabling it as a default column for the campaign.
The Number of Opportunities column will display how many opportunities that the lead currently has. You can click the icon to view a breakdown of the opportunities by won, lost and pending. By default the Number of Opportunities column is not displayed, it can be shown using the column options or by enabling it as a default column for the campaign.