What exactly is a floating IP and how does it work?

I am setting up a high availability load balancer. I found this tutorial from Digital Ocean which says that two load balancers can share a common floating IP.

https://docs.digitalocean.com/products/networking/floating-ips/#how-to-implement-an-ha-setup

What is this floating IP and how does it work?

Can one set it up by one's self (without using Digital Ocean)

to answer your question:

DigitalOcean describes a Floating IP as

a publicly-accessible static IP address that can be assigned to one of your Droplet.

So basically it is an IP within Digital oceans' infrastructure that you can assign to your resources, so they are reachable from the public internet, and I do not believe these can be set-up without digital ocean, since they are part of Digital Ocean Network fabric.