What are the types of interactions in biological network (protein networks)?
Most of them are pretty self-explanatory. KEGG pathways are often vague, as a lot of the specific biology for some proteins has yet to be properly investigated - sometimes you only know that two proteins associate, not what they actually do. 

The relationship between the two is easier to predict when you know the function of one, e.g. a protein interacting with a kinase is likely phosphorylated by said kinase. A lot of those relationships correlate to post-translational modifications (glycosylation, phosphorylation, ubiquitination, acetylation, methylation, etc) made to one protein by another. Wikipedia actually has a decent overview of these with lots of examples.

Several of the other relationships have to do with regulating protein activity or expression. This goes somewhat hand in hand with what I stated above. Some proteins are only activated once phosphorylated or demethylated, etc. Other enzymes may remove phospho groups, leading to repression. Others still may interact directly with a protein, inhibiting it from binding its substrate or target. Maybe the binding induces a conformational change in one protein that allows it to bind its substrate when it can't normally. Some proteins function as transcription factors that activate or repress the expression of certain genes, indirectly affecting protein levels.

There are all sorts of possibilities - the KGML files try to represent them in a way that at least gives some indication of how the interactions between two proteins may affect each. I hope that provides some insight.