A Ligase is an enzyme that can ligate ("glue together") two molecules with a covalent bond, using energy from ATP.
DNA ligase can ligate broken DNA strands, forming covalent phosphodiester bonds.
An example of how a ligase works (with sticky ends):
5'-AGTCTGATCTGACT GATGCGTATGCTAGTGCT-3' 3'-TCAGACTAGACTGACTACG CATACGATCACGA-5' becomes 5'-AGTCTGATCTGACTGATGCGTATGCTAGTGCT-3' 3'-TCAGACTAGACTGACTACGCATACGATCACGA-5'