Background check is done by processing application where your education and experience credentials are from - this task verifies your claims and authenticity of documents.
Background check is verification of your name, date of birth and country you claim to be from. If its application for citizenship background check is first thing they do about addresses, places of work and verification with info of the database in CRA and Service Canada and basic residency calculation.
Security check is done by Canadian intelligence and security agencies, with or without cooperation of proper authorities in countries you lived/stayed in, and it is done completely independently from IRCC - IRCC only gets result, which will be positive or negative, without any explanation (as it is classified). This task looks deep into your life(LinkedIn Facebook), activities and associations to find out if you are/were involved or are/were associated in any way with anyone who is/was involved in espionage, terrorism, organized crime, radical movements, crimes against humanity, war crimes, etc..
Criminal check looks into your police certificates and verifies with relevant countries to see if you have a criminal record that may make you inadmissible to Canada.
In short,
Criminality and Security together are background checks. Criminality check is performed by RCMP, and Security by CSIS. Eligibility just means checking whether you're qualified to become a citizen (tax filings, physical presence, language proof etc.)