The UCLA site explains Cronbach's alpha as the average internal correlation among survey items. It also says that it is not a measure of unidimensionality. Rather, it is a measurement of internal consistency (though just intuitively I feel what is coherent tends to be also uni-dimensional... I think the point is that the measure is most optimal by design for the assessment of internal correlation, not dimentionality.
Standardized versus Raw
This SAS website says one should use the standardized version of the measure (as opposed to raw).
It says: "Because the variances of some variables vary widely, you should use the standardized score to estimate reliability."
A note to myself: Does this mean if I standardized all items before the analysis, I get the same value for raw and standardized? I can experiment this.