Equity and debt are two broad classification of sources of finance for any firm, these are not mutually exclusive. It is not as if a firm must choose only one of the two type of sources of finance. A firm may choose to employ both equity and debt in different proportion.
Also, usually every firm starts with financing from equity, and uses debt to supplement equity, rather than replace equity. It is quite rare to find firms with zero equity finance.
Then it is important to note that within the broad classification of equity and debt financing many different alternative form of finance exist. For example equity finance includes alternatives like, direct money invested in proprietorship or partnership firms or issue of shares.
The issue of equity shares also presents different choices such as form of company (private limited, public limited, cooperative,, etc.) and type of shares (ordinary, preference, etc.). Similarly debt financing can also take many forms such as supplier credit, short term bank loans, long term loans from banks or other institutions, fixed deposits from the public, and debentures.