Let me try and explain.
Firstly, the EU basic regulation (218/2008) and the related 'Part 21' for airworthiness of aircraft, supported by rules drafted by EASA and approved by the Commission and the political process, embrace prima facie all aircraft registered in EU Member States. The exceptions to this scope are in Annex II to 218/2008 and cover aircraft such as microlights (< 450kgs MTOM), historic aircraft (pre-1955 design / pre-1975 production), home-built aircraft, ex-military aircraft, hang-gliders etc. Annex II aircraft remain under the airworthiness jurisdiction of each respective Member State and their own rules.
Thus in the UK, an Annex II aircraft will still have a UK CAA C of A which requires renewal every 3 years, together with associated maintenance and compliance with ADs (Airworthiness Directives) etc. In the case of certain categories of aircraft this process may look different, as with Permit to Fly aircraft (non-C of A) but the essential principles are the same.
For aircraft subject to EU regulation, again the system looks different but the fundamental principles are the same. An aircraft requires a C of A - thus it is a 'certified aircraft' compliant with the original airworthiness requirements including the relevant design code and production process etc. The first difference is that unlike the old (but still current for Annex II aircraft) UK system the C of A is not an expiring document - it is for the life of the aircraft. But, the validity of the C of A is subject to periodically meeting the other requirements, the main one of which is an annual review of the aircraft's continued airworthiness state over time. This review is documented in a process leading to the issue of an ARC - an (annual) Airworthiness Review Certifcate. The process that underpins this requires an assurance that all relevant ADs etc have been complied with (during the last year), that any mandatory modifications carried out since the last ARC have been implemented in accordance with offical data etc and that the aircraft's maintenance has been carried out according to the legal requirements, which usually means the manufacturer's maintenance manual. This maintenance programme is quite separate from the ARC process itself and is often on a different timing to the due date of the ARC, though it is convenient to try and align the dates.
Within the EU / EASA laid-down processes there is now a clear separation of functions between those who can issue an ARC and those who can conduct the maintenance and release the aircraft to service. This is not the case with the old CAA system (which still applies to Annex II aircraft). The former is a 'sub part G' organisation known as a CAMO - Continuing Airworthiness Management Organisation, whilst the latter is a sub part F organisation.
Depending on the aircraft category / class some of the maintenance can be performed by the owner / operator (see Annex VIII of Part M) whilst more complex maintenance has to be done by qualified experts / engineers etc. Within Part M (= Maintenance) there is the option to go for a 'controlled environment' whereby one contracts the work to an approved organisation, or the 'uncontrolled environment' whereby the owner can 'shop around' more freqeuently. That's a whole new discussion, but hopefully the above means that you are a little wiser!