Suppose I have a corpus of sentences, which I then use a part of speech tagger to annotate.
What algorithms or techniques can I then input the tagged data into? Are the tagged words somehow useful just by themselves?
The first use to come to mind is syntactic parsing. If you have a rule based grammar where your symbols are POS tags, then you could generate the tree representation of the sentence. Or, if you have a statistical tree parser, then it can use the POS tags as features for determining the tree/parse structure.
In my own experience in NLP in both industry and academic research, I've primarily used POS tags for machine learning features.
They often are valuable features in Named Entity Recognition tasks and other ontology labeling tasks.