Peptides are the family of molecules formed from the linking, in a defined order, of various amino acids. The link between one amino acid residue and the next is an amide bond, and is sometimes referred to as a peptide bond. An amide bond is somewhat shorter than a typical carbon-nitrogen single bond, and has a partial double-bond character, because the participating carbon molecule is doubly bonded to an oxygen molecule and the nitrogen has a lone pair of electrons available for bonding.
Peptides (like proteins) occur in nature and are responsible for a wide array of functions, many of which are not yet understood.
Peptides differ from proteins, which are also long chains of amino acids, by virtue of their size. Traditionally, those peptide chains that are short enough to make synthetically from the constituent amino acids are called peptides rather than proteins. The dividing line is at approximately 50 amino acids in length, since naturally-occurring proteins tend, at their smallest, to be hundreds of residues long.
Notes on terminology: