What is called the "area size that an animal usually lives in" or "needs for a normal life"? Is there any specific term?
Are you looking for 'Home range' (see also the definition in Encyclopaedia Britannica)? Generally, 'home range' is defined as the entire area an individual animal uses, while the 'territory' is the subset of the home range that is actually defended from conspecifics (in animals that show territoriality). 'Home range' is often delimited by the types of environments used and movement patterns of individuals.
To avoid confusion I just want to add to fileunderwater's answer the equivalent words we use to describe the "area size a population/species lives in".
The spatial range a single individual occupies is generally called
home range or
territory (as fileunderwater said before me).
The spatial range a single species (or population) occupies is called
geographic range or
species range and we call the boundaries of this range the
geographic range limits or sometimes called
range limits only.
Perhaps also Habitat.
The term typically refers to the zone in which the organism lives and where it can find food, shelter, protection and mates for reproduction...