Typically,
A borough is an area under control of a local authority, such as a council. Local government. Political.
A district is an area with a purpose, such as a shopping district or a financial district.
A neighbourhood is an area where people live. A community, a collection of houses. Residential.
Emerald_Pick•
The other commenters answered your question, but since we're on an English Learning sub: "explain me" should be "explain to me." (Ideally: "explain the difference … to me.") You can use "teach" in both ways, but "explain me" means something similar to "What am I?"
Fit-Share-284•
A district is a well-defined area that has a specific purpose (e.g. residential districts, industrial districts, etc.). A neighbourhood is a community with resential housing and/or businesses and shops. A borough is a British term for neighbourhood (I might be wrong, but I don't hear it a lot in Canada). In any case, the three words are close synonyms and are interchangeable a lot of the time.