The Ainur were all created outside of Arda. It is unknown if other Ainur were created outside of Arda afterwards, but no being of similar essence was created on Arda.
The beings who existed before Arda are called the Ainur. They were created by Eru Ilúvatar.
There was Eru, the One, who in Arda is called Ilúvatar; and he made first the Ainur, the Holy Ones, that were the offspring of his thought, and they were with him before aught else was made.
The Silmarillion, Ainulindalë, The Music of the Ainur, p. 3
After Arda was created, some Ainur descended upon it: the Valar and the Maiar.
The Great among these spirits the Elves name the Valar, the Powers of Arda, and Men have often called them gods. [...] With the Valar came other spirits whose being also began before the World, of the same order as the Valar but of less degree. These are the Maiar, the people of the Valar, and their servants and helpers.
The Silmarillion, Valaquenta, p. 15
Note that the Valar and Maiar are both Ainur. Counted among the Maiar are the Balrogs, the Istari (the five "Wizards") and Sauron.
Eäla and fëa
The distinction between the Ainur and any other being is that their spirit is disincarnate: they do not require a body to survive. Melkor, however, would eventually become permanently incarnate.
But fair and noble as were the forms in which they were manifest to the Children of Ilúvatar, they were but a veil upon their beauty and their power.
ibid., p. 20
Thus, there are two different kinds of spirits. The essence of the Ainur is an eäla, a disincarnate spirit. Other beings are made of two parts: a fëa ("soul") and a hröa ("body").
These were the (eälar) spirits who first adhered to [Melkor] in the days of his splendour, and became most like him in his corruption: their hearts were of fire, but they were cloaked in darkness, and terror went before them; they had whips of flame. Balrogs they were named by the Noldor in later days.
There is a footnote to the word eälar in this passage:
'spirit' (not incarnate, which was fëa, S[indarin] fae). eäla 'being'.
The History of Middle-Earth Volume 10: Morgoth's Ring, The Later Quenta Silmarillion (I), p. 165
Only the Ainur have an eäla and no Ainur were ever born on Arda.
The only union between an Ainu and another being was between Melian, a Maia, and Elu Thingol, an Elf. Their only child was Lúthien, an Elf who later became mortal out of her love for Beren.
Ungoliant was said to have had many children, one of which was Shelob. Her exact nature is unclear, and it is unknown whether she was a Maia or some other sort of spirit. Therefore, the nature of her children is also unknown.
If she were a Maia, we could speculate that, as with Melian and Lúthien, her children might have been of a different essence.
In very early versions of The Lost Tales, a precursor to The Silmarillion, the Valar did have children on Arda:
With [Manwë] was Varda the Beautiful, and she became his spouse and is Queen of the Stars, and their children were Fionwe-Urion and Erinti most lovely.
The History of Middle-Earth Volume 1: The Book of Lost Tales, Part One, The Music of the Ainur, p. 58
Even Melko had a son with the ogress Fluithuin: Gothmog, lord of Balrogs:
Tuor stood then in the way of that beast, but was sundered from Egalmoth, and they pressed him backward even to the centre of the square nigh the fountain. There he became weary from the strangling heat and was beaten down by a great demon, even Gothmog lord of Balrogs, son of Melko.
The History of Middle-Earth Volume 2: The Book of Lost Tales, Part Two, The Fall of Gondolin, p. 183
The relationship between Yavanna, Aulë, and Oromë was also somewhat strange:
[...] and Oromë was born of Yavanna, who is after named, but he is not Aulë's son.
The History of Middle-Earth Volume 5: The Lost Road and Other Writings, The Later Annals of Valinor, p. 110
All of this would eventually disappear and the concept of the Maiar would replace it. Since at this point the concepts of eälar and fëar were nonexistent, it is also unclear whether these "Children of the Valar" were also Ainur.
One of the last mentions of the Children of the Valar was this:
[...] in the margin against these changes my father wrote:
Note that 'spouse' meant only an 'association'. The Valar had no bodies, but could assume shapes. After the coming of the Eldar they most often used shapes of 'human' form, though taller (not gigantic) and more magnificent.
At the same time the passage concerning the Valarindi, the Children of the Valar, at the end of §4 was struck out (as it was also on the top copy), since this note is a most definitive statement that any such conception was out of the question.
The History of Middle-Earth Volume 10: Morgoth's Ring, The Annals of Aman, p. 69