How are evolution and variants of aircraft considered by certification authorities? Recently, both Boeing and Airbus launched "new" aircraft as a modernized version of their respective best sellers (B737MAX and A320neo). To narrow the question down, I will consider the A320 and the FAA, but the same goes for the B737 and almost certainly many more aircraft families, and I hope for other certification authorities as well. I'm not really clear on the difference between type, model and series but I think the difference may be relevant for the certification process. Perhaps highlighting this difference is enough to answer the question.

Wikipedia lists multiple variants of the A318. Do the A318-111 and the A318-121 require different certifications?

Moreover, when adding a variant to the family (here adding the A318 to the family composed of the A319, the A320, and the A321), do the certification authorities consider this aircraft as a totally new one or only as a variant of existing one (many shared components)? Can this consideration lighten the certification process?

Is the new version (A320neo) considered as a new aircraft by the certification authority and does it thus require the same set of tests as a new aircraft?

In short, when a new aircraft is an evolution or a variant of a previous one, how is it considered by certification authorities?
Variants are seen as a modification of an existing airplane design, and only the changed parts need to be certified. The real advantage is that the set of rules the certification has to comply with are frozen at the time of the initial application, so any additional rules that had been added later are not relevant. This is called "grandfather rules".

This definitely lightens the certification process and makes it less complex, but makes the safety enhancements, which were added after the type was certified first, optional for newly built airplanes. 

The A318, A319 and A321 are all variants of the original A320, and the subtypes you name differ in their engine. Each engine option has to be certified separately, but all certification work not related to the engine can be shared between the variants. Additionally, the purchaser will select his options for instrumentation and furnishing, chosen from a limited set of previously certified options.

When a new version is introduced, the public will get a very different message from the airplane maker than the certification authorities. For the PR, the new variant is all new, but the FAA and the EASA will hear that almost nothing changed. In the end, it is similar to corporate profits: The shareholders get told the company made oodles of money, while the tax authorities will be presented with a bleak picture of mainly losses.