I have visited some sources and I can't finally understand that Eta (η) in Ancient Greek pronounced like 'eɪ' (delay), or like 'eə' (hair). In fact, should we say an 'ɪ' at the end of pronouncing η, or it is pronounced only like long 'e'?
As Asteroides has said, the "classical" pronunciation (from the fifth century BCE or thereabouts) is reconstructed as
/ɛː/, a longer version of the vowel in English "red".
However, there are quite a lot of different ways to pronounce Ancient Greek—and unlike with Latin, the "reconstructed classical" version hasn't caught on as thoroughly. One of the more prominent systems is the "Erasmian" system, which was originally based on the works of Erasmus but has evolved in all sorts of different ways in different places and different traditions.
In the traditional English "Erasmian" system, Ε is pronounced as in "red" (
/ɛ/), and Η and ΕΙ both as in "raid" (
/ej/). This isn't especially accurate to how Plato and Aristophanes would have spoken, but is not uncommon among modern scholars.
(Another system pronounces it all like Modern Greek, so Ε and ΑΙ are
/e/ and Η, ΕΙ, Ι, Υ, ΟΙ, ΥΙ are all
/i/ as in "reed". This is arguably more accurate than the "Erasmian" system, because it's how the Greek language actually evolved over time, but can make it difficult to distinguish words because so many vowels merged together.)
Sources that say Ancient Greek eta was pronounced like the vowel in English "delay" or "hair" are only providing a loose approximation of the vowel.
Greek η was always a monophthong
The pronunciation changed over time. The standard reconstruction for Classical Attic Greek η is [ɛː], a long front mid-low monophthong. In modern Greek, the vowel has been raised to [i] (a front high monophthong), and between [ɛː] and [i], we reconstruct the intermediate stage [eː] or [e] (a front mid or mid-high monophthong) for Koine Greek.
For more information on Greek pronunciation and the timeline of changes such as [ɛː] > [eː~e] > [i], see the answers to Resources for pronouncing Greek.
There is no evidence that I know of for η ever being pronounced as a diphthong by Greek speakers.
English speakers often diphthongize vowels
In some accents of English spoken in southern Britain, hair is pronounced with a long monophthong [ɛː] (the symbol "eə" is a non-phonetic conservatism in transcriptions of these accents). For a speaker of this kind of English accent, the hair vowel is a close approximation to the oldest reconstructed pronunciation.
However, in many other dialects of English, the contrast between short [e] and long [ɛː] does not come very naturally, and cannot easily be described by reference to the pronunciation of English words like "late", "delay", or "hair". Since most English speakers who learn ancient Greek do so for the purpose of reading texts, and none are learning for the purpose of speaking to Ancient Greek speakers (which is obviously not possible), many pronunciation systems in common use by English speakers don't emphasize accuracy to the phonetic reconstruction of Ancient Greek. Some pronunciation choices or recommendations that you find in texts may instead made primarily for the purpose of making it easier for English speakers to remember the spelling of Greek words.
For example, Mitchell L. Holley writes in "Greek Pronunciation: The Pedagogical Pertinence":
The Erasmian system possesses great heuristic value for the new Greek student. By using the system of Erasmus, students learn a pronunciation system that allows them to quickly memorize the spelling of words, which is a vital part of learning an inflected language.
For the sake of this goal, it doesn't matter whether an English-speaking student distinguishes epsilon and eta as [e] and [ɛː], [e] and [eə], [ɛ] and [ɛː], or even [ɛ] and [eɪ]. But there is no reason to think that Ancient Greek speakers shared English speakers' tendency to diphthongize long vowels.