I have been asked to check for pseudonodes - none are allowed for streams but they are for roads. I can't figure out why so I assume I am not understanding what a pseudonode is. I thought it was a node with no reason to be there. I am using ArcGIS 10.0.
A psuedonode is a point at which two nodes overlap unnecessarily. You're on the right track with your understanding but not quite there. It's not that the node itself has no reason to be there, it's that a pair of nodes exists where there should only be one. For example, and what I think you're supposed to be checking for, where what should be a single line feature is split into two or more distinct lines (each possibly containing multiple vertices).
In your case, the rules are basically saying that all streams should begin and end either at a vertex of another stream which is not an endpoint (node), or at a vertex that is an endpoint not shared by any other lines. Interpretation of that rule can be a little tricky. If the segments on both sides of the candidate pseudonode share all the same attributes, then it is a psuedonode. If one attribute changes, like width, it may or may not be depending on other rules (like width can only change at a confluence). If you have a confluence of two streams with a third leaving that point, if all have different names or widths then it would not be a psuedonode - whereas if widths are all the same and the name is on the downstream and one of the upstream, it would be a pseudonode but only between two of the segments, not the third.
Roads are often broken for other reasons, or it may just be that for the purposes of your exercise they only want you to worry about the errors in just the stream layer.
There are some graphics on this page that may help futher explain. There is also this handy pdf poster that graphically explains various topology rules (one of which is pseudonodes). Also you may want to read about the distinction between nodes and vertices.