Shareholder's Equity consists of two main things: The initial capitalization of the company (when the shares were first sold, plus extra share issues) and retained earnings, which is the amount of money the company has made over and above capitalization, which has not been re-distributed back to shareholders.
So yes, it is the firm's total equity financing-- the initial capitalization is the equity that was put into the company when it was founded plus subsequent increases in equity due to share issues, and retained earnings is the increase in equity that has occurred since then which has not yet been re-distributed to shareholders (though it belongs to them, as the residual claimants).
Both accounts are credited when they increase, because they represent an increase in cash, that is debited, so in order to make credits = debits they must be credits. (It doesn't mean that the company has that much cash on hand, as the cash will likely be re-invested).
Shareholder's Equity is neither an asset nor a liability: it is used to purchase assets and to reduce liabilities, and is simply a measure of assets minus liabilities that is necessary to make the accounting equation balance:
Shareholder's Equity = Total Assets - Total Liabilities
Since the book value of stocks doesn't change that often (because it represents the price the company sold it for, not the current value on the stock market, and would therefore only change when there were new share issues), almost all changes in total assets or in total liabilities are reflected in Retained Earnings.