Someone I talked to used two dots in this word:
I thought that it's my screen having dirt on it, but the dots in fact are part of the glyph. Can someone explain that?
The origin of "naive" is the French word "naïve". (Notice that the French "naïve" is italicized)
As a French word, it is spelled naïve or naïf. (French adjectives have grammatical gender; naïf is used with masculine nouns while naive is used with feminine nouns.) The two dots above the "i" are called diaeresis. As an unitalicized English word, "naive" is now the more usual spelling, although "naïve" is unidiomatic rather than incorrect.
For more, look up this.
Naive is borrowed from from which in turn borrowed the umlaut from the German language. The umlaut is used in vowels to indicate that they are pronounced as two vowels combined. For example ä is pronunced like "ae". You then know that naive with the double dots (naïve), is not pronounced as nāve, but rather as nī•ēv like the way we know it.
Source, Naive explanation