relationship -A "relationship" segue is the segue between a container view controller and its child or children -- so, the initial controller of a navigation controller, the view controllers in the tabs of a tab bar controller, and the master and detail controllers of a split view controller.
Show - Pushes the destination view controller onto the navigation stack, moving the source view controller out of the way (destination slides overtop from right to left), providing a back button to navigate back to the source - on all devices
Example: Navigating inboxes/folders in Mail
Show Detail - Replaces the detail/secondary view controller when in a UISplitViewController with no ability to navigate back to the previous view controller
Example: In Mail on iPad in landscape, tapping an email in the sidebar replaces the view controller on the right to show the new email
Present Modally - Presents a view controller in
various different ways as defined by the Presentation option, covering
up the previous view controller - most commonly used to present a view
controller that animates up from the bottom and covers the entire screen
on iPhone, but on iPad it's common to present it as a centered box
overtop that darkens the underlying view controller and also animates up
from the bottom
Example: Tapping the + button in Calendar on iPhone
Popover Presentation - When run on iPad, the
destination appears in a small popover, and tapping anywhere outside of
this popover will dismiss it. On iPhone, popovers are supported as well
but by default if it performs a Popover Presentation segue, it will
present the destination view controller modally over the full screen.
Example: Tapping the + button in Calendar on iPad (or iPhone,
realizing it is converted to a full screen presentation as opposed to an
actual popover)
embed - An "embed" segue is the segue between a container view and the controller that's embedded in that container view that you get automatically when you add a container view to a controller's view.
Custom - You may implement your own custom segue and have complete control over its appearance and transition.
Note that you may see these segues termed as Selection Segues, Manual Segues, or Adaptive Segues. The deprecated segues are not adaptive for various devices. These segue types are deprecated in iOS 8: Push, Modal, Popover, Replace. They're basically the non-adaptive equivalent of those described above.
Check out Building Adaptive Apps with UIKit, Session 216, that Apple presented at WWDC 2014. They talked about how you can build adaptive apps using these new Adaptive Segues, and they built a demo project that utilizes these segues.
Both of these segues are executed as soon as the parent controller gets instantiated. You do not call them, but you can implement prepareForSegue, and pass information to the destination view controller.