It simply means after and before with respect to the flow of information in a given pathway. For example, consider this schematic representation of a gene expression pathway:
TF1 ==activates==> gene1 ==produces==> Kinase1 ==phosphorylates==> ProtA ==produces==> Prot1
In this schema, Kinase1 is downstream of TF1 and upstream of ProtA. Or, to take a classic example (source):
MEK1/2 is downstream of Raf and upstream of ERK1/2.