An amide is a compound containing the functional group RnE(O)xNR'2 (where R/R' can be hydrogens or various other organic groups).
Peptide, on the other hand, refers to a short chain of amino acids in which are linked by amide groups (which, the case of peptides, can be referred to as peptide bonds).
An amide bond is a peptide bond. However, amide is a term used in the context of small molecules while peptide is used in the context of polymers (like peptides and proteins).