UI = User interface, HCI = Human-computer interaction, HMI = Human-machine interaction.
UI is a field of HMI/HCI it’s the thing that user uses to interact with computer/machine. With UI user can do input to give command to computer or machine like a car, which would be turning a wheel for example or hitting a brake and computer/machine will provide output based on your input, like in code if I type: Console.WriteLine(“Hello World!”); and run it computer will put output: “Hello World!” to console or if I hit a brake car will start to slow down for example or even when I’m opening a door with doorknob I’m using an interface.
Usually with UI people mean graphical interface of a program or lets say a game. 
Like in a game if you click spell icon from interface character does that spell for example.
In desingning an interface for a game for example you need to take in account a few things. It needs to be simple for people to fast learn and use it, and it helps if it is customable so the player/user can custom it the way he likes. Then when you have done your interface let people test it, then analyze the data and do the fixes needed and repeat untill it’s good.
HCI means interaction between user and computer and studying, designing, implementing and developing new interfaces or ways to interact with computer. The ultimate goal of HCI is to minimize barrier between user and computer.
HMI is almost same as HCI, but it’s a lot older word and it’s related to machines aswell, like lets say cars. In car you choose direction it goes with wheel and you choose your speed with accelerator and brake. In HMI desing the goals are same as in HCI: make machine easier to use, and reduce barrier with machine and user.
So ultimately UI is just a way to tell computer/sofware/machine/game what you want it to do, and when you desing interface you want it to be as easy to use as possible.
Ati Marin 