The follow-up selection affects when you get to call back the lead and how you'll be notified about it.
Selecting after will simply postpone the lead and put it back in the top of the queue at the time of follow-up.
Selecting in exactly will schedule the lead and prioritize it in the top of the queue when it is due, and send you an email notification* within a couple of minutes after it becomes due if you are not already looking at it. Leads scheduled for an exact followup is listed under the "Scheduled" submenu.
* In order to receive email notifications about due leads, you need to enable those in your notification preferences (enabled by default).
We recommend choosing in exactly for leads that you agreed on calling at a specific time and after for the rest.
Note : By default, we ignore follow-up times for anything but call backs. If you want to set a follow-up time for a winner or loser (which can be useful if they are later migrated), you must select a follow-up that is different from the default. Archived leads are never set for follow-up.
The relative options — Next week, Week after next, Next month, Next quarter and Next year — land on the first day of that period: the day your week starts, the 1st of next month, the 1st of the next quarter. Weeks follow the "Week starts on" setting in your profile, so they begin on Sunday if that is how you have it set. The remaining options ("14 days", "1 hour" and so on) are counted from the moment you pick them. Every option shows the date it resolves to, so you can see where a follow-up lands before you choose it, and the list is ordered by that date.
The time of day comes from your own default call back time, so if you set that to 7 these follow-ups all arrive at 07:00.
Pro Tip: You can customise default call back times from your preferences. Click the user icon to the upper right and choose "My Preferences", then "Working".