I'm quite sure that the process is managed by the states. There have been relatively recent instances (in the last 20-30 years) when particular Southern states were found to be in violation of Federal Law and had to submit to monitoring and add corrections, but my memory is that this oversight was found to be no longer justified by US Supreme Court rulings (in the last 10 years). {Wish I could be more vague :-)}
The State of California's voting registration is managed by the California Secretary of State (an elected office). Voting can be either by mail or in person. The in-person election process is handled by the counties. The Site Map for Elections and Voting Information: http://www.sos.ca.gov/elections/site-map/
The Legislative authority and directions: http://leginfo.legislature.ca.gov/faces/codes_displayexpandedbranch.xhtml?tocCode=ELEC&division=2.&title=&part=&chapter=&article=
Current Regulations: http://www.sos.ca.gov/administration/regulations/current-regulations/
The Sec'y of State maintains an online mechanism to check a persons current status in its system: https://voterstatus.sos.ca.gov/
There was a recent discovery the registration process at California Department of Motor Vehicles had allowed thousands of ineligble voters to be added to the registration rolls kept by the State. NYT article: https://www.nytimes.com/2018/10/10/us/california-dmv-voter-registration.html
Here's the DMV website and notice that it includes quite a bit of information about registration in general: https://www.dmv.org/ca-california/voter-registration.php
2018 Poll Worker Training Standards: http://www.sos.ca.gov/elections/voting-resources/voting-california/help-strengthen-our-democracy/poll-worker-info/poll-worker-training-standards/
Poll Worker Training Guidelines 2005 (PDF version): http://elections.cdn.sos.ca.gov/poll_worker_training_guidelines_final_draft1.pdf
Poll Worker Information: http://www.sos.ca.gov/elections/poll-worker-information/