The "base" portion of Thymine contrasted with the backbone.
RNA and DNA are polymer chains of nucleotides. Nucleotides are often referred to as bases, which is short for nucleobase.
The base is the unique part of each nucleotide. It's similar to the sidechain of an amino acid. The backbone portion of each base is the same, although the backbone of DNA is different that the backbone of RNA, and both are different than protein backbone.
The "base" in "nucleobase" indicates the group is chemically basic (as opposed to acidic), but this fact isn't particularly important when working with DNA and RNA.
See DNA Gallery and RNA Gallery for more.