Companies Act 2013 replaces The Companies Act
Public Limited Companies or Limited Companies ( Section 2 (71). Its usually defined as a Company which is not a private Company.
Private Limited Companies ( Defined under Section 2 (68) ) 
One Person Company
Section 8 Companies
Producer CompaniesPublic Limited Companies
Public Limited Companies can have more than 200 shareholders. Shares are freely transferable. However compliance requirements are lot greater than private Companies. Public Companies can be Listed (on stock exchanges) or Unlisted.  For Unlisted public Companies, the Corporate Identification number (CIN) starts with 'C' and for Listed Companies, it starts with 'L'.
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