I have observed that the word naïve is written with two dots on the i. Why is this? Is it correct to write the word with a single dot, as in naive? Are there any other English words with such two dots?
It's called a dieresis. It's used to show that the "a" and the "i" are not to be pronounced as a single sound. So it's pronounced something like "na-eve" and not like "knave" or with the "ai" rhyming with the "i" in "knives".
But in 50 years as a native English speaker/writer, I have never written it like that, and have rarely seen it so either.
Another example is "cooperative" where the second "o" in theory has a dieresis. It's pronounced "coh-op...." and not to rhyme with "loop". Again, I've never written it with the dieresis, and don't recall seeing it like that either.
The two dots on the letter i are a French diacritic sign. The two dots in the French spelling naïf/naïve show that ai has not its normal pronunciation but is spoken as two separate vowels /a-i/. In English you can write naive or naïve.
The French term for the two dots on e/i/u is tréma.
The Greek term diaeresis means separation and refers to the separate pronunciation of two succeeding vowel letters.
Basically the answer is that naïve is sometimes spelled with the diaresis because it is derived from French which spells it that way. It is actually very uncommon for native English speakers to spell it with the diaresis, largely because, as you've noticed, the diaresis is not normally a part of the English language. The vast majority of English keyboards don't even contain a modifier to add a diaresis (or a tilde, accent, or any other marking, for that matter) to a letter. However, the auto-correct feature in some computer programs will change naive to naïve, as my browser has done in this post.
In some cases in English, the two dots indicate an umlaut, typically seen on loan-words (predominantly from languages like German and Swedish), to indicate a special pronunciation of the vowel:
ångström, Bön, doppelgänger, filmjölk, föhn wind, fräulein, Führer, gemütlichkeit, glögg, Gewürztraminer, Götterdämmerung, Gräfenberg spot, jäger, kümmel, pölsa, smörgåsbord, smörgåstårta, über, Übermensch, surströmming...
As others have stated, however, this is not why we find it in words like naïve. For this class of words, the symbol is not an umlaut but a diaresis (or diæresis). For these, it is to mark a vowel as being unassociated with another vowel, either adjacent as in naïf, or elsewhere in the word, as in Brontë. This class of words includes both loan-words (particularly from Romance languages: naïveté), and home-grown English terms (reënter).
Boötes, Brontë, caïquejee, Chloë, continuüm (rare), coöperate [-ion, -ive], coöpt, coördinate [-ed, -ing, -ion, -or, -ors], daïs, faïence, Laocoön, naïf, naïve, naïveté, Noël, noöne (rare), oöcyte, oölogy (rare), opïum (rare), öre, preëminent [-ly] (rare), preëmpt [-ion, -ive] (rare), reëlect [-ed, -ing] (rare), reënter [-ed, -ing] (rare), reëstablish [-ed, -ing] (rare), residuüm, spermatozoön, Zaïre, Zoë, zoölogy
Especially now in the days of the keyboard, both forms of this diacritic tend to be omitted for simplicity when writing or printing English. The only words that appear to have any extra resulting ambiguity from homographs are Öre, Bootës and Coöp.
The New Yorker style guide is the only one in common use in the US which still advocates their use: for most people, both umlauts and diaresis are considered as archaic as digraphic ligatures (æ and œ).
The purpose of it is to show that the word is two syllables, and that the i falls into each. Think of the two dots as being a sort of divide, so the two syllables are "nai" and "ive" rather than the i only belonging in one of them (na-ive or nai-ve).
Another example is the word "weird". While most of us would pronounce it as "weerd" this isn't the case in Scotland. As with naive, the word has evolved - to a greater extent - to exclude the use of two dots (Shakespeare always spells weird with both, so it definitely used to be that way). This gave the word two syllables, "wei" and "ird".
After the Revolution, America became determined to shorten words to simplify them as much as possible (part of the reason the letter u was removed from words like "colour" and why "z" often replaced "s"), which is why over there "weird" is always said quite short. In other places, such as England and Australia, the word is often still drawn out very slightly so it almost has a second syllable. This is a remnant of when it was always pronounced with two syllables - the second syllable is still very prominent with a Scottish accent because of how the r is rolled.
The adjective "naïf" (or naïve which is the feminine writing ) is a French word. The pb is that French language "marries" some vowels together to produce another sound. Normally, a+i makes a [e] like in navy: the a of navy equals the a+i in French among other ways of writing that sound. to prevent it, there can be either an H between the a and the i or there will be what is called a "trema", the 2 points above the letter i to indicate that the a and the i are pronounced separately. I hope this clarification will be helpful. Kind regards.