A carboxyl group in its standard form consists of a carbon, two oxygens, with a hydrogen attached to one of the oxygens. The entire group is attached to another atom, usually carbon. This form is known as carboxylic acid, indicated by formula -COOH.
In Foldit, the carboxyls are shown in their charged state, with the hydrogen missing. This form occurs in the aqueous solution inside a cell, and has the fomula -COO. This form has a negative charge.
The carboxyl group in an amino acid backbone is attached to the alpha carbon.
Two amino acids, aspartate and glutamate, have a carboxyl group at the end of their sidechains. They're also known as "aspartic acid", and "glutamic acid", since the carboxyl makes their sidechains acid under some conditions.
In aspartate, the sidechain carboxyl is attached to the beta carbon. In glutamate, the beta carbon is followed by the gamma carbon, and the carboxyl group is attached to the gamma carbon.
A carboxyl is one type of carbonyl group, so the term "carbonyl carbon" may appear in places, referring to the "C" in -COOH or -COO.