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Required:

PS4 System
PS Vita with firmware version 3.00 or later

Optional:

Remote Play- or Second Screen- compatible game (Games that require a camera or Move controller are not compatible)
Wireless Router (To connect via home wi-fi network)
Wireless Broadband Internet Connection (To connect via Internet. 3G, 4G, and LTE are not supported)

Types of Remote Play Connections

The PS4 can connect to the PS Vita in three different ways. The PS4 Link app on the PS Vita will pick the connection method best suited to your current situation.

PS4 as Wireless Access Point: This option lets your PS Vita wirelessly connect directly to the PS4 and provides the best Remote Play experience.

Home Wi-Fi Network: The PS Vita wirelessly connects to a router, which in turn connects to the PS4. Use this option if your home router is closer than your PS4 to where you'd like to Remote Play.

Wireless Internet Connection: With this method, the PS Vita connects to the Internet, then to an Internet-connected PS4. Use this option if you're away from home and want to connect to your PS4.
How to Connect and Start Remote Play

On the PS4:

Go to (Settings) > [PS Vita Connection Settings] and make sure the [Enable Remote Play] box is checked.
From the same menu, ensure the [Connect Directly to the PS Vita] box is checked. This will provide the best Remote Play experience. If you're experiencing issues with a direct connection, you can un-check this box to bypass the wireless access point option and connect via your home wi-fi network.

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On the PS Vita:

Tap (PS4 Link) > [Start] and choose the feature you'd like to use (Remote Play or Second Screen)
(If you've previously registered with a PS4, you will not need to perform this step.) If this is the first time you've connected to the PS4, you'll need to enter the 8-digit number displayed on the PS4's screen, and then tap [Register]. You have 5 minutes to enter this code. If you are unable to do so, simply tap [Back] on the PS Vita and you'll receive a new code.
The PS Vita will search for your PS4 and connect via the best connection method. You will see a notification on your PS4 stating that a user has connected via Remote Play.

The PS Vita's screen will now display the PS4's screen if you are using Remote Play.

If you are using the Second Screen feature the screen will display the appropriate Second Screen visuals, or if your game isn't currently using the feature the PS Vita's screen will read "This screen is currently not in use."


Credit's:
Sony For Some of the wording
TD-Pictures