Both twenties and 1925 are nouns; periods of time, specifically. And they are also both objects of prepositions: in and after, respectively.
There are two general rules of English you can follow to figure out that these are nouns. First, the definition of noun: person, place, or thing, generally speaking. Dates and times are things. Second, the use of prepositions, which require a noun target. In what? The twenties. After what? 1925.
Is it a common noun or proper noun?
These are both common nouns. Twenties refers to a general group of numbers with two as the tens digit. Because it doesn't refer to anything specific, it cannot be a proper noun.
Even though 1925 refers to a specific year, it's still a common noun, because it's a number. Numbers just aren't proper nouns in English, even though there's only one specific year in this case, because 1925 can be used in a broad range of contexts and mean subtly different things, depending on the context. A year, an age, a count, the result of a calculation, etc.