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I heard about many use: Bitcents, Bitmills (or Bitmilles)

1.02 - One bitcoin and two bitcents
0.001 - One millibitcoin (nickname "one Millie")
0.000001 - One microbitcoin (nickname "one Mike")
0.00000001 - The bitcoin base unit (nickname "one Satoshi")

millicoins, microcoins, and nanocoins

Which are the official naming conventions and should Bitcoin be capitalized when you are referring to the unit?

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"Official" units of bitcoin

To my knowledge, only the bitcoin and the satoshi are "officially" named units of bitcoin.
As of 2014/2015, one also often sees bit as a moniker for one microbitcoin, i.e. 100 satoshis.

BTC has been used as the abbreviation for bitcoin for a long time, yet, in September bitcoin was added to ISO 4217, as XBT, where X means not associated with a country, just as in XAG (Silver) and XAU (Gold), and opposed to national currencies, e.g. USD (US Dollar). However, it seems to me that XBT has not been adopted by the community, BTC remains prevalent.

Origin of millibitcoin and microbitcoin

Millibitcoin and microbitcoin follow the conventions of the International System of Units, and thus are derived from the main unit bitcoin with a prepended "m" for millibitcoin and a prepended "μ" for microbitcoin.
Following the same conventions and starting from the common abbreviation BTC for bitcoin, one would use mBTC and μBTC.

Millibit and microbit are confusing

While usually people will understand the meaning, using the abbreviations "millibit" or "microbit" gives the impression that they are assiociated with bit as the main unit. This is especially confusing with the advent of bit as a nickname for microbitcoins.

"bit"

Bit is a terrible name for two reasons: Bit, the binary digit, is the basic unit of information. It is used a lot in the context of anything related with computers and therefore shouldn't be doubled as a subunit of bitcoin. Second, due to people using "millibit" and "microbit" it is terribly confusing to anyone that is new to Bitcoin, but familiar with the International System of Units.

Millie and Mike

"Millie" and "Mike" were proposed nicknames for millibitcoin and microbitcoin, but I haven't seen them get much usage. I don't think that they will see adoption.

Units in English

In general currencies are not capitalized in English, unlike the name of the protocol. So, you would write about the Bitcoin protocol, but five bitcoins.

Overview

I would argue for the following usage of units:

Unit       Value in bitcoins  Value in satoshis   Abbreviation
-------------------------------------------------------------------
bitcoin        1.0               100,000,000             BTC
millibitcoin   0.001                 100,000            mBTC
microbitcoin   0.000001                  100            μBTC
bit            0.000001                  100             bit
satoshi        0.00000001                  1         satoshi

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