What are the names of the various versions of the Android OS, and how are these names chosen?

Many people refer to Android versions by "codenames" rather than version numbers (e.g. Android "Gingerbread"). What are the names of the various versions of the Android OS, and how are these names chosen?

Each version of Android since 1.5 has been developed with a specific codename. These codenames are chosen alphabetically, and have thus far all been dessert items (or, generically, sweet/sugary foods). Some codenames are associated with more than one version number, while others are limited to only a specific one, and the reason for this inconsistency is not currently known. The naming typically appears to correspond to changes in the developer API levels, but this is not always true (example: 3.0 and 3.1 are both "Honeycomb" but they have different API levels).
The following names are used for the currently existing Android releases. Note that versions 1.0 and 1.1 were not publicly named. However, Android 1.1 was internally referred to as "Petit-Four".

Cupcake:

Android 1.5

Donut:

• Android 1.6
  
  
  

Eclair:

• Android 2.0
• Android 2.1
  

Froyo: (short for "frozen yogurt")

• Android 2.2/2.3
  

Gingerbread:

• Android 2.3
  

Honeycomb:

• Android 3.0
• Android 3.1
• Android 3.2

Ice Cream Sandwich:

• Android 4.0

Jelly Bean:

• Android 4.1
• Android 4.2
• Android 4.3
  

KitKat:

• Android 4.4
  

Lollipop:

• Android 5.0
• Android 5.1
  

Marshmallow:

• Android 6.0