A monomer is a protein consisting of a single chain (or polypeptide). (The word derives from ancient Greek, mono- "one" and -mer "part".)
Many Foldit design puzzles have "monomer" in their titles. These puzzles involve designing a single monomer without reference to anything else.
In contrast, symmetry puzzles with "symmetric design" in their titles involve multiple identical copies of a monomer. These puzzles still involve designing a monomer, but the monomer must also form hydrogen bonds to the other copies.
The proteins in symmetry puzzles are referred to as oligomers ("few parts"), and include dimers, with two copies of a monomer, trimers with three copies, tetramers, with four copies, and so on.
In some discussions, the designable main chain may be referred to as "the monomer" to distinguish it from the symmetric chains, which mirror the design of the main chain.