A rapporteur is responsable for a certain topic on behalf of a committee in the European Parliament. Each political group may then have a shadow rapporteur to negotiate the topic with the rapporteur. Every group has also a coordinator for each committee.
Rapporteurs, shadow rapporteurs and coordinators often coordinate and decide the outcome of the political deliberations. The President and the up to 4 Vice Presidents in each committee have a more formal and less political role.
The power are mainly with the coordinators. The actual composition of coordinators can be found at the websites for the different committees.
The role of coordinators, rapporteurs and shadow rapporteurs are inserted in the rules of Parliament after proposal from a Reform Working Group in the EP.