If you lodge a protest, complaint or appeal etc. then you make an official or a formal protest, complaint or appeal etc.
Lodge can be used interchangeably with the term “bring”. So, you could say “the landowner brought and action against his neighbor for trespass.” What is meant in either case is that some formal statement has been filed with a court or another organization in a manner such that it must be processed by that court or organisation.
Courts and organization have formal requirements regarding the different kinds of complaints filed with them. In many instances forms are available and instructions must be followed. Also, at least in the judicial context, filing fees must be paid. Once all the requirements are fulfilled and the court or organization has accepted the filing, the complaint is considered “lodged” with the court or organization.