To use an android, will it run on Wifi ? How does it work?

Yes - assuming you HAVE a Wifi router connected to the internet (i.,e. like one in your own home or a friend's house that you know the Wifi network name and password/WEP of).

Still, it depends what you want to do.

Normal phone calls made with the phone's built-in dialer require a cellular contract. You cannot make those over Wifi with most phone. However, if you call using an app that does voice calls over IP, then you can use Wifi to make calls.

Normal texting, with the phone's built-in messaging system requires a cellular voice contract (data contract may be required for any file/picture attachments and for group messaging). If you need to text over Wifi there is a way to do that via email but you need to know both the recipient's phone number and what cell towers they use, and what email address that cell company uses for sending texts. For example, if a person has a Tracfone, and you know THEIR tracfone uses Verizon, and their phone number is 123-456-7890, then to email a text to them the address would be 1234567890@vtext.com (I know because I have a Verizon-using phone and I looked it up).

For getting on the internet, Wifi can be used. I actually keep my cell data turned off so I don't use it accidentally and I use my phone to access Twitter and the general internet with Wifi whenever I can.

Yes, you can run wi-fi. I have setup a wi-fi only phone for my little sister and it works for what she needs.

LTE data is nice because it takes the internet to go

Almost all Android devices can use Wi-Fi. You don't need Wi-Fi to use it, though it becomes limited in what you can do without Internet.