Company

There is one statue called Companies Act, 1956. Broadly companies and corporations, both name styles are governed under the provisions of this Act - unless an entity is formed by a Special Act of the legislature; for example Food Corporation of India. Basically both are companies but the corporation status is perceived mainly in relation with 100% government companies or subsidiaries of government companies or government, its agencies, its authorities or its institutions being the major shareholders. Additionally, the Registrar of Companies, guided by its administrative guidelines or in his own prudent judgement may grant the use of the word "corporation" in the name of the company ( to a private party) having complied with certain formalities; examples being Indo-Gulf Corporation Limited - a Birla Group company and Indian Oil Corporation Limited - a majority capital with the Central Government . 

In modern times, both the entities are mainly limited liability companies and governed by the provisions of the Companies Act, 1956 or the provisions of the Special Act, as the case may be. 

Companies, ending their names with "Private Limited" (or its abbreviations like P. Ltd or Pvt. Ltd.) or "Limited" (or Ltd.) are limited liability private and public limited companies respectively with limited liability clause for its shareholders. There are other kind of companies also permissible to be registered under the Companies Act, 1956 - like Section 25 companies etc. 

With this background, the difference between a company or a corporation may be of cosmetic nature so far as the ownership and internal workings are concerned. 

The word corporate is a general word used both for company and corporation, but in legal sense a corporate entity must be a registered entity either under the Companies Act or under the Special Act as enumerated above but excludes registered societies in all forms, partnership and proprietary firms, trusts, AOPs, HUFs etc...

hope its clear now...