Learn Romanian: Am cumpărat un telefon folosit.

Just like in English, Romanian frequently uses the past participle of a verb as an adjective. Here, folosit is the past participle of a folosi (to use). Because it is acting as an adjective, it must agree with the noun telefon in gender and number.

Telefon is a neuter noun. How does that affect the ending of folosit?

In Romanian, neuter nouns act like masculine nouns in the singular and feminine nouns in the plural. Since telefon is singular here, the adjective takes the standard masculine/neuter singular form folosit. If you bought multiple used phones, it would switch to the feminine plural form: telefoane folosite.

Is folosit the most natural way to say "second-hand" in everyday Romanian?

It is perfectly natural and correct, but Romanians also have other very common ways to express this. You will often hear the English phrase second-hand used as an invariable adjective, or the native idiom la mâna a doua (literally "at the second hand").