In this site, I see a variety of acronymic names for C.elegans neurons but what do these names mean (for example AVAL, AVAR)?
Unfortunately, there doesn't seem to be a very systematic nomenclature. Some of the letters are acronyms while others describe a class of neurons. The letters V and D usually describe the ventral and dorsal counterpart of a pair of neurons. The same goes for L and R (left and right).
This list names all C. elegans neurons and you can see that, for example, the "ring interneurons" or motor neurons start with RI or RM. "Inner labial" or "outer labial" neurons are called IL or OL. Ventral cord motor neurons start with a V and their different classes are called VA, VB, VC and VD plus a number. The classes are determined by lineage origins or historic reasons ;) And there are a lot of exceptions to these rules, because their historic names remain even though we continue to learn more about these neurons.
The example AVAL/AVAR are ventral cord interneurons. They form a class with other ventral cord interneurons called AVB, AVD, AVE. So, the first two letters refer to the main class, the third letter to the subclass. The L or R refer to the left and right member of this pair.
Some friends working in C. elegans neuroscience told me that you basically just treat these letter codes as individual names for the neurons without bothering about their origin and meaning too much.