In our club, prospective members are proposed through a member completing a "Membership Proposal and Application" and submitting it to the Club Services Director. The form and supporting documentation can be downloaded here. The board will discuss the proposal at the next opportunity, and the proposing member will be informed of the decision. Membership is not to be discussed with the candidate until the board has made its decision.
Once approved, new members must complete a personal data page for our records. It is available here. Useful information for new Rotarians (and perhaps many long-standing ones!) is available here.