All I want to know about these words is their meaning difference and how they are used in sentences... (If you can give me examples it would be great :))
Your question is both very easy and very hard. To the extent that there are differences they are very subtle.
- District. As in Pittsburgh’s Strip District, a strip of wholesale and specialist stores along the Allegheny River, with an diverse population
- Quarter Like the Latin Quarter (Quartier Latin) on the left bank of the Seine, known for its intellectual and literary interest, close as it is to the University of the Sorbonne and the School of Fine Art (Beaux Arts)
- Neighbourhood applies to any relatively self-contained portion of a town or city, usually with housing rather than blocks of flats (at least in the UK).
- Village as in Greenwhich Village in New York, very artsy and, in its day, alternative culture.
- Town. as in Camden Town in London, which was once just a town in the traditional sense before being absorbed into London, but now more associated with young people and families, outdoor markets and antiques. It could have been called ‘Camden Village’. Camden is also a huge London borough and could easily be described as a ‘district, but not really a neighbourhood.
I am afraid that travel is the only way to become confident in your use of these overlapping concepts. But what can be said about your two words is that the word district, unlike the word quarter, can be used as an official or administrative subdivision of a city or even large town. ‘Quarter is more unofficial (though you can find it on maps because it designates an area tha is know to be interesting and fun to visit.