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ASCII is a 7-bit code - one bit (binary digit) is a single switch that can be on or off, zero or one. Character sets used today in the US are generally 8-bit sets with 256 different characters, effectively doubling the ASCII set.
Each character is encoded as 1 to 4 bytes. The first 128 Unicode code points are encoded as 1 byte in UTF-8. These code points are the same as those in ASCII CCSID 367. Any other character is encoded with more than 1 byte in UTF-8.
In Java, a character is encoded in UTF-16 which uses 2 bytes, while a normal C string is more or less just a bunch of bytes.
It depends what is the character and what encoding it is in:
An ASCII character in 8-bit ASCII encoding is 8 bits (1 byte), though it can fit in 7 bits.
An ISO-8895-1 character in ISO-8859-1 encoding is 8 bits (1 byte).
A Unicode character in UTF-8 encoding is between 8 bits (1 byte) and 32 bits (4 bytes).
A Unicode character in UTF-16 encoding is between 16 (2 bytes) and 32 bits (4 bytes), though most of the common characters take 16 bits. This is the encoding used by Windows internally.
A Unicode character in UTF-32 encoding is always 32 bits (4 bytes).
An ASCII character in UTF-8 is 8 bits (1 byte), and in UTF-16 - 16 bits.
The additional (non-ASCII) characters in ISO-8895-1 (0xA0-0xFF) would take 16 bits in UTF-8 and UTF-16.
That would mean that there are between 0.03125 and 0.125 characters in a bit.
There are 8 bits in a byte (normally speaking in Windows).
However, if you are dealing with characters, it will depend on the charset/encoding. Unicode character can be 2 or 4 bytes, so that would be 16 or 32 bits, whereas Windows-1252 sometimes incorrectly called ANSI is only 1 bytes so 8 bits.
In Asian version of Windows and some others, the entire system runs in double-byte, so a character is 16 bits.
EDITED
Per Matteo's comment, all contemporary versions of Windows use 16-bits internally per character.
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