Be sure to read the 'Wireless' page for general information before you read this page.
If you choose to use Explorer with a WiFi connection you should read this note about a possible bug.
To use the WiFi option, you need a specific WiFi dongle that is compatible with the EV3 brick. Currently there is only one model available that works with an unmodified EV3: "Netgear WNA1100" (available from Amazon) which must be inserted into the large USB connector at the left side of the EV3. If you don't want this rather large dongle to stick out from the brick, then make it stick up vertically instead with a cheap right-angle adaptor such as this one (FR, UK).
Once you have successfully established a connection between the EV3 and your router, the EV3 should have received an IP address, which would look something like "10.2.3.4". Find the IP address that was assigned to your EV3 in the "Brick Info" section of the settings menu and make a note of it. It is at the very bottom of the list.
Important note: once you have established a wireless connection with the brick (whether it be via Bluetooth or WiFi) you must then close all settings dialogs on the brick before attempting to run programs via the wireless connection. If any settings dialog or any brick application such as 'port view' is running on the brick then it will be impossible to launch any EV3 Basic program, wirelessly or not, on the brick from either Small Basic or EV3 Explorer. It will, however, still be possible to use other functions of EV3 Explorer such as compiling programs to the brick.
When you want to finally connect to the EV3 from a Small Basic program or with the EV3-Explorer, use the "WiFi" button in the dialog box that asks for which connection to use. Then enter the IP address exactly as it was shown on the brick. You only have to enter the address once. For future uses, it will already appear in the selection list.
Note that the brick does not automatically connect to the WiFi router when starting up. You always have to activate the connection after each power-up.