Melkor was originally the most powerful of the Valar. He fell to evil and became known as Morgoth. He gradually lost his power, but he never became a Maia - the chief difference between a Vala and a Maia is not in power, but in status. The Valar were the rulers and the first ones to enter the World. The Maiar were their followers and servants, and they came to the World later.
Sauron was Morgoth's most powerful servant, and his most trusted lieutenant. He was a Maia. This does not mean that he was a Balrog, though - not all of Morgoth's Maiarin servants were Balrogs. The word 'Balrog' means 'Demon of Might' and refers only to those spirits of fire who followed Morgoth.
Sauron was a Maia who followed Morgoth, but he wasn't a fire-spirit. Therefore he was not a Balrog.
Arien was a spirit of fire and a Maia, but she didn't follow Morgoth. Therefore she was not a Balrog.
Gothmog was a Maia who followed Morgoth, and he was a fire-spirit. Therefore he was a Balrog.
'Elf' - 'Elves'
'Vala' - 'Valar'
'Maia' - 'Maiar'
I hope this clears up most of the confusion.
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