POS tagging would give a POS tag to each and every word in the input sentence.
Parsing the sentence (using the stanford pcfg for example) would convert the sentence into a tree whose leaves will hold POS tags (which correspond to words in the sentence), but the rest of the tree would tell you how exactly these these words are joining together to make the overall sentence. For example an adjective and a noun might combine to be a ‘Noun Phrase’, which might combine with another adjective to form another Noun Phrase (e.g. quick brown fox) (the exact way the pieces combine depends on the parser in question).
You can see how parser output looks like at http://nlp.stanford.edu:8080/parser/index.jsp
A shallow parser or ‘chunker’ comes somewhere in between these two. A plain POS tagger is really fast but does not give you enough information and a full blown parser is slow and gives you too much. A POS tagger can be thought of as a parser which only returns the bottom-most tier of the parse tree to you. A chunker might be thought of as a parser that returns some other tier of the parse tree to you instead. Sometimes you just need to know that a bunch of words together form a Noun Phrase but don’t care about the sub-structure of the tree within those words (i.e. which words are adjectives, determiners, nouns, etc and how do they combine). In such cases you can use a chunker to get exactly the information you need instead of wasting time generating the full parse tree for the sentence.