In molecular biology, what's the meaning of the terms "downstream signaling" and "upstream signaling"? What's the difference between them?
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 pathway:
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.
If you think of a transcription factor, than this factor needs to be regulated. This happens either by signalling pathways and also on the gene expression level (how much of the factor is available). So everything what regulates this factor is called upstream (look at the image below).
Everything which is regulated by our factor (other genes) is located downstream of it. I think these terms come from the flow diagrams which are pretty usual in this context.
Look at the figure (from SABiosciences)
Here the signal comes from the membrane and is mediated through different factors until its passed into the nucleus.