In some journals, such as Nature, it is compulsory, where you can specifically define the contribution of each author. The corresponding author is usually the person who is able to answer any scientific question about the paper, including the experimental detail and also the one who will be responsible of any possible mistake. This is, in most cases, the first author because he/she is also the one submitting the article and, therefore, establishing the communication with the editorial board.
There is no standardised method to quantify contributions. Some journals require you to list which component activities each co-author participated in (e.g. writing manuscript/experimental work/data analysis/etc). Going beyond such a qualitative assessment to quantifying contributions further is not really practical.
There is an interesting correspondence on 'Author Sequence and Credit for Contributions in Multiauthored Publications' . Authors have made an interesting Comparison of the Credit for Contributions to This Paper under the Four Different Models Suggested in the Text.