During translation ribosomes decode the genetic information present in the mRNA and protein synthesis takes place. During this process which of those two does move, the ribosome or the mRNA?
Here's an example in which the ribosome is fixed:
During co-translational translocation, the ribosome is essentially anchored onto the ER membrane through the Sec61 complex. It certainly cannot move along the mRNA. The mRNA is fed through the ribosome and the nascent peptide traverses into the ER lumen.
The ribosome moves relative to the mRNA by, in effect, pulling itself along it. If both the ribosome and the mRNA are freely floating and not attached to anything else (as in jp89's answer), the relative amount of movement should depend on their relative masses.
(Actually, it also depends on how much drag each of them experiences with respect to the surrounding liquid medium, but since I have no idea how much that is, and since it's probably highly conformation-dependent anyway, I'm going to just ignore that and just assume that the drag is also more or less proportional to mass, at least to first order.)
As it happens, a quick Google search and some back of the envelope calculation suggests that the mass of a ribosome and the average mass of an mRNA are both around a megadalton. Of course, the length (and thus the mass) of an mRNA varies quite a lot, so it would seem likely that sometimes it's the ribosome that moves mostly, sometimes it's the mRNA, and sometimes it's both.
Also, as shigeta and others have pointed out, there can be more than one ribosome attached to the same mRNA strand. That's going to make the mRNA move more (and, correspondingly, the ribosomes move less), since there are more ribosomes pulling it along. Then there's also the protein being transcribed, which is attached to the ribosome but also being moved with respect to it. And I really have no idea how negligible the interactions with the tRNAs and so on are. It's a mess, but my guess would be that, usually, it's mostly the mRNA that moves, but that the ribosomes aren't completely stationary either (unless they're attached to something, of course).
Ps. Here's an exercise for you, which you may try out if you happen to have a friend who works at a public swimming pool. Otherwise consider it a gedankenexperiment. You know those floating ropes that separate the lanes in the pool? Try getting your friend to let you into the pool when it's not in use and to release one of the ropes from the walls. Then get in, grab the rope with your arms and legs and try pulling yourself along it. While doing so, try to decide whether it's you or the rope who moves more. (Also, to more closely approximate the Reynolds numbers involved inside a cell, imagine doing this in treacle instead of water.)
Movement is relative. The real events happening in translation are the conformational changes of ribosome makes itself continuous reading the base sequentially. Please refer the biochemistry textbook or cell biology textbook.
Indeed, if the ribosome is anchored, you may say the mRNA is moving.
The mRNA moves during translation. It is essentially threaded through the ribosome. This has been known ever since polyribosomes were discovered; see paper here. Polyribosomes are a cluster of ribosomes that read a series of mRNA molecules. Often, the ribosomes in a polyribosome will be translating the same mRNA.