What are the common productivity measures of a scientist, like h-index?
The h-index is common (and the g-index, which corrects for self-citation), as is the Journal Impact Factor. Johan Bollen has a good review of the various metrics.

However, it's important to point out that all those measures are just different ways of counting citations. They don't account for things like code you've written or talks you've given and they can't address systematic bias in citation practices such as coercive citation or citation mutation. Also, any citation-counting metric will penalize younger researchers simply due to the time it takes to publish one paper and for other papers to get published citing yours. In order to keep academics from having to publish a paper just to describe some code they've written or a dataset they've accumulated, aggregators have been built to pull in these various metrics and consolidate them. Total Impact is a good example of such a system. The general field of study looking at incorporating these broader metrics is called #altmetrics, and you can find a collection of research on the topic here.