Once you know how to use Handoff, it's very simple and can be quite useful. The second method is from the app switcher, where any application waiting to be handed off will appear to the left of the homescreen.
It should be noted that if you have a passcode or Touch ID enabled you will still need to authenticate before it takes you to the app. Swiping upward will take you to the spot in the application open on your other device. The first is via the lock screen, where an icon will appear in the bottom left corner of the display. Going from OS X to iOS or from one iOS device to another has two ways of handing off an application. I don't have multiple devices to test this, but I would assume a second icon appears if two iOS devices are close and have open applications that support Handoff. As you can see above, when transitioning from an iOS device to a Mac an icon will appear on the side of the dock which will show the currently active application on the iOS device if it supports Handoff. The way Handoff works depends on which direction you're going. As of right now the list is as follows, but it will continue to expand with both Apple's applications and third party ones from developers implementing Handoff into apps that have versions for both OS X and iOS. It allows you to start tasks on one device and continue them on another, provided the application supports it.
Handoff is the big star of the show when it comes to leveraging the power of an ecosystem and controlling the software on every device you sell. Like with Airdrop, the iPad 3 is left out when it comes to Handoff support, but it does support SMS and Call Forwarding.
Apple has also disabled Handoff on the 2011 MacBook Airs and 2011 Mac Mini despite their support for Bluetooth 4.0. iOS users will need an iPhone 5 or later, iPad 4th generation / Mini 1 or later, or the iPod Touch 5th generation. The heavily connected continuity features like Handoff and Airdrop will require Bluetooth 4.0 support which limits devices on the OS X side to Macs released in the last few years. What Apple users need to know about continuity is they need to have two or more supported devices. The topic of platform lock-in is a complicated one and could spawn a whole article in itself, so I won't go into detail with it. Unfortunately, users who only have a single device or have products spread out across many companies and operating systems are unable to take advantage of these sorts of features. Going forward, the power of a product and service ecosystem is going to be a key point of differentiation between companies that make their own software and those that piggyback off of another platform. Apple has always had some degree of integration and commonality, but continuity is really the first time Apple has really taken advantage of the ecosystems they have created and brought customers into. Continuity is essentially an umbrella term for all the features that allow iOS and OS X devices to work together. By far the most exciting features that Apple has created this year are part of continuity.