A device that has built-in Wi-Fi has an internal adapter that is able to receive and connect to available wireless networks. If the network requires a password, it must still be entered in order for the device to connect.
Wi-Fi refers to a wireless network that becomes available when in range of a router. A modem modulates and demodulates the analog signal from a cable or telephone line into data, essentially connecting the attached devices to the Internet. A router takes this data and transforms it into a radio signal that broadcasts within a vicinity. This is known as a Wi-Fi network. Devices that are able to connect to Wi-Fi networks, such as ones with built-in Wi-Fi, can convert the signal back into data and communicate with the router, allowing it to access the Internet without being attached to the modem.
Some things to look out for when looking at devices with built-in Wi-Fi is the difference between "built-in Wi-Fi" and "Wi-Fi ready." Devices that have built-in Wi-Fi can connect to a network out of the box, but devices that are Wi-Fi ready require a separate adapter to do so. The ready part merely means it has a port for an adapter, which is an important thing to keep in mind when selecting a product.