Four different ways to display math can all be used. They must be used for equations of any significant length as these forms can be line broken under relevant transformations. Hence, the visually identical use of centre, single dollar and \displaymath is not structurally the same and should not be used.
You may label numbered equations; you cannot use \tag, \notag (AMS) or change the equation counter in any way except that required above or the equation numbering in the various formats will not match (though there would be internal document consistency in each of the formats).
Do not put full line comments in display math mode and do not finish with a \\ as both can result in problems with transformation.
All displaymath environments are capable of being reflowed in output formats that are reflow capable. The line breaks that are specified will always be retained but additional line break will occur as necessary. Hence, it is important to enclose in braces any subpart of the equation that must not be broken. Line breaks are most likely to occur at relations and binary operators, they can occur within brackets. In some reflow algorithms the algorithm will attempt to align relations. Hence, if you are encoding a condition e.g.
x = y \mbox{if} y = 0
the condition should be in braces to prevent this i.e.
x = y \mbox{if} {y=0}.
Line breaks cannot occur within mbox or text blocks. Line breaks occur at commas or whitespace but could in some formats in the future. It is important to consider whether your equations have adequate line breaking opportunities. If not you will need to line break the equation by hand for use at a larger font size.
\begin{equation}\label{equation} \sum_{i=1}^{15} x_i^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 + x_8^2 + x_9^2 + x_{10}^2 + x_{11}^2 + x_{12}^2 + x_{13}^2 + x_{14}^2 + x_{15}^2 \end{equation} \begin{displaymath} x_1^2 = x_2^2 = x_3^2 = x_4^2 = x_5^2 = x_6^2 = x_7^2 = x_8^2 = x_9^2 = x_{10}^2 = x_{11}^2 = x_{12}^2 = x_{13}^2 = x_{14}^2 = x_{15}^2 \end{displaymath} \[ \prod_{i=1}^{15} x_i^2 = x_1^2\ x_2^2\ x_3^2\ x_4^2\ x_5^2\ \ x_6^2\ x_7^2\ x_8^2\ x_9^2\ x_{10}^2\ x_{11}^2\ x_{12}^2\ x_{13}^2\ x_{14}^2\ x_{15}^2\ \] $$ \prod_{i=1}^{15} x_i^2 = x_1^2 \cdot x_2^2 \cdot x_3^2 \cdot x_4^2 \cdot x_5^2 \cdot x_6^2 \cdot x_7^2 \cdot x_8^2 \cdot x_9^2 \cdot x_{10}^2 \cdot x_{11}^2 \cdot x_{12}^2 \cdot x_{13}^2 \cdot x_{14}^2 \cdot x_{15}^2 $$\begin{equation} \label{equation} \sum _{i=1}^{15} x_ i^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 + x_8^2 + x_9^2 + x_{10}^2 + x_{11}^2 + x_{12}^2 + x_{13}^2 + x_{14}^2 + x_{15}^2 \end{equation}\[ x_1^2 = x_2^2 = x_3^2 = x_4^2 = x_5^2 = x_6^2 = x_7^2 = x_8^2 = x_9^2 = x_{10}^2 = x_{11}^2 = x_{12}^2 = x_{13}^2 = x_{14}^2 = x_{15}^2 \]\[ \prod _{i=1}^{15} x_ i^2 = x_1^2\ x_2^2\ x_3^2\ x_4^2\ x_5^2 \ x_6^2\ x_7^2\ x_8^2\ x_9^2\ x_{10}^2\ x_{11}^2 x_{12}^2\ x_{13}^2\ x_{14}^2\ x_{15}^2 \]\[ \prod _{i=1}^{15} x_ i^2 = x_1^2 \cdot x_2^2 \cdot x_3^2 \cdot x_4^2 \cdot x_5^2 \cdot x_6^2 \cdot x_7^2 \cdot x_8^2 \cdot x_9^2 \cdot x_{10}^2 \cdot x_{11}^2 \cdot x_{12}^2 \cdot x_{13}^2 \cdot x_{14}^2 \cdot x_{15}^2 \]
One of the forms above is a numbered equation 1 and can be labelled and referenced via \ref, \eqref cannot be used.
You can use the eqnarray and eqnarray* environments. Currently one of the transformations won’t number the numbered version of this environment. This is a very recently reported bug in software which is being actively developed. It may well be fixed by the time you need to use it.
As noted above your linebreaks will be honoured but your alignment might not be in some transformations. Extra linebreaks might be inserted! See comments above on how to work in this environment.
You must put in both ampersands on each line or some transformations will not function — specifically in office formats the below always transforms to an array-like structure with 3 columns. If the columns aren’t all present in the original this causes significant layout problems.
Text can be placed in eqnarray etc. however line breaking cannot break \mbox so the text must be as short as possible.
\begin{eqnarray}\label{eqnarrayref} \sum_{i=1}^{13} 2^i &=& 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10} + 2^{11} + 2^{12} + 2^{13}\nonumber\\ &=&2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 + 4096 + 8192 \nonumber\\ &=&16382 \qquad\mbox{text in formulas does not break} \end{eqnarray}\begin{eqnarray} \label{eqnarrayref} \sum _{i=1}^{13} 2^ i & =& 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10} + 2^{11} + 2^{12} + 2^{13}\nonumber \\ & =& 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 + 4096 + 8192 \nonumber \\ & =& 16382 \qquad \mbox{text in formulas does not break} \end{eqnarray}
Here is a reference to the above equation (2).