The powers and responsibilities (as well as identities) were fixed in the Golden Bull of 1356 under Charles IV. of the Luxembourgs. Although generally not much of it was new this was the first time that the established traditions, understandings and agreements were written down in a single proper document.

Besides setting a fixed protocol for the election process they had quite a few neat privileges. For example they were allowed to collect tolls and mint their own money.
But I think most electors' favourite parts were on one hand the forced primogenitur, meaning the electoral lands would always be inherited in full by a single heir instead of getting split up between eligible parties, and on the other hand far-reaching legal immunity.

This immunity is primarily found in the privilegium de non appellando illimitatum, which granted all electors the unlimited right to forbid their subjects from appealing an electorate court's decision at the Supreme Courts of the Holy Roman Empire (the Reichskammergericht and the Reichshofrat).
Coupled with that is also the privilegium de non evocando, which let them forbid their subjects from primarily calling on the Imperial court and skipping the local one entirely (although some protected classes could always do so regardless); this was later made a general principle with the introduction of the Reichskammergericht in the big reform of 1495.

And of course one should not fail to mention the political power that came with those privileges and the fact that the Emperor and everyone who was interested in becoming his successor had to be best chums with them, especially during the traditional pre-electional round of bribery and political concessions.