Not all rugby players have the same footwear needs, and if you've never played before (or even if you have and you maybe wonder where all the blisters are coming from), you need to understand this to play your best and to be as safe as possible.
Let's start with what not to wear. It is illegal for you to wear boots made for American football unless you cut off or otherwise remove that cleat in the middle of the toe (see photo). That particular cleat is illegal because, unlike in American football, there is a good chance you will get your face stepped on in rugby, and there is a good chance that particular cleat could end up in your eye socket should the person stepping on you be wearing it.
So unmodified football boots are illegal. Other athletic shoes, like sneakers/trainers/plimsouls, are merely unwise: you need some sort of cleat for traction even if you're playing on artificial turf (which is getting more and more common). Also, anyone who wears gym shoes/whatever-you-call-them on the pitch is likely to be targeted for having their toes stomped on. This, again, is another physical risk of the sport (hey: noone said you have to play), and some types of boots actually provide some protection - and we'll talk about them in a second - but running shoes do not. At all.
So, assuming I haven't scared you off, let's go through what types of boots you should use for different situations.