The meaning of those words depends a lot on the administrative system in the country you are asking about. You probably should specify which country you are asking about to get a more specific answer. Here - boroughs used to be semi-autonomous parts of a city. They effectively don't exist any more, apart from historical references. A ward is an electoral area of a city - a councillor will typically be elected to represent a ward. We don't have districts here as such, although 'district' is often used in the name of a local body to show that it is a rural area, rather than an urban area.

Not a complete answer, but just FYI The only common use of borough that I know of (being a native English speaker from America) are the five boroughs of New York City.

that's quite an interesting question, borough refers to self governing towns and wards are subdivision of a local area which is usually used to subjugate electoral purposes and district refers to administrative divisions of areas that are governed by local government hence you'd have district wards which are various wards within a district borough is quite a rare thing now, and also, wikipedia is your best friend!