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A canonical variate is a new variable (variate) formed by making a linear combination of two or more variates (variables) from a data set. A linear combination of variables is the same as a weighted sum of variables. Because we can in infinitely many ways choose combinations of weights between variables in a data set, there are also infinitely many canonical variates possible.
In general additional constraints should be satisfied by the weights to get a meaningful canonical variate. For example, in canonical correlation analysis a data set is split up into two parts, a dependent and an independent part. In both parts we can form a canonical variate and we choose weights that maximize the correlation between these canonical variates (there is an algorithm that calculates these weights).
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