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End of proof symbol. The "\end{proof}" command generates a "qed symbol" (by default, a hollow square) at the end of the proof. The symbol can be redefined with "\qedsymbol"; in particular, if you don't want qed symbols, just redefine "\qedsymbol" to an empty command as follows: "\renewcommand{\qedsymbol}{}". If this definition is placed in the preamble, its effect is global. However, one can also place the definition inside a "\begin{proof} ... \end{proof}" pair. In this case, it only affects the current proof and eliminates the qed symbol in that proof, but has no effect on other proofs.
Placing of qed symbol. An important rule is that you should not leave a blank line before "\end{proof}" since that would indicate a paragraph break and would cause the qed symbol to be placed one line below where it should be.
Proofs ending with an equation. If the proof ends with a displayed equation, then "\end{proof}" would normally place the symbol one line below the display, which looks odd. To place the symbol on the same line as the display, add "\qedhere" at the end of the display (right before "\end{equation}", or its equivalent). Note that in order for this to work properly, the display should not be set with double dollar signs (which isn't advisable anyway) - use \[, \] or one of the display environments instead.
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