Contigs are long continuous stretch of sequence results from overlapping short reads.

When you do a paired-end sequence, you sequence a DNA fragment ( lets say of length 400bp ) from both the ends, resulting R1 and R2 reads. When you map them back to genome, if the both reads map with in 400bp (as they are generated from 400bp fragment) and expected strand orientation, they are referred as concordant pairs. This is primarily in DNA-Seq. I don't think there is a discordant mate pair, it is called discordant pair. Mate-Pair sequencing is a different thing.

When one of the mates (either R1 or R2) is not present (lost due to pre-processing or in mapping step or post-processing of data), the other mate becomes single (also sometimes referred as orphan reads) called as singletons.