What are the types of NRI Bank Accounts non-residents can open?

Non-residents can open Demat or NRE or NRO Account in Indian Banks.
Demat Account for NRIs:
In the advanced countries, depository systems and services have played a significant role in not only facilitating smooth online share trading and settlement but also attracting foreign investment in capital market. This is exactly what the National Securities Depositories Limited (NSDL) plans to do in India. The depository system as evolved by the NSDL enables investors to overcome all problems related to physical certificates. The Securities and Exchange Board of India (SEBI) regulate the functions of NSDL.
NRE and NRO Accounts:

Non-residents can maintain two types of Rupee Account – Non-Resident (External) Rupee Accounts (NRE Accounts) and Ordinary Non-Resident Rupee Accounts (NRO Accounts). The difference between NRE and NRO Account lies in that the funds remitted from abroad or local funds, which can otherwise be remitted abroad to the account holder, can be credited to NRE Accounts. Local funds, which do not qualify for remittance outside India, are required to be credited to NRO accounts. It may be noted that money may be transferred from NRE to NRO Account but not vice-versa, and money once credited to NRO Account cannot be transferred back to NRE Account as the money becomes non-repatriable. Apart from all this, NRIs can use these accounts to make investment in Indian mutual funds.