Map Reduce is a inner component of hadoop, other Pig and hive are hadoop eco systems it means run on the top of hadoop. The purpose of both mapreduce, pig and hive purpose is process the vast amount of data in different manner.
Mapreduce: apache implemented it. highly recommendable to process entire data, it's time consume and required program skills like java (highly recommendable), pyghon, ruby and other programming languages. total data aggregate and sort by using mapper and reducer functions. Hadoop use it by default.
Hive: Facebook implemented it. most of the analysts especially bigdata analysts use this tool to analyze the data especially structure data. Backend this hive tool use mapreduce to be processed. Internally Hive use special language called HQL, It's subset of SQL language. Who is wellever in SQL, they can goes with Hive. It's highly recommended to the Datawarehouse oriented projects. Much difficult to process un structured especially schema-less data.
Pig:
Pig is a scripting language, implemented by Yahoo. The main difference between pig and Hive is pig can process any type of data, either structured or unstructured data. It means it's highly recommendable for streaming data like satellite generated data, live events, schema-less data etc. Pig first load the data later programmer write a program depends on data to make it structured. Who is expert in programming languages they will choose this Hadoop ecosystems.