MapReduce is a parallel programming technique made popular by Google. It is used for processing very large amounts of data. Such processing can be completed in a reasonable amount of time only by distributing the work to multiple machines in parallel. Each machine processes a small subset of the data. Check out the following article for an introduction to MapReduce.
Hadoop is a framework written in Java for running applications on large clusters of commodity hardware and incorporates features similar to those of the Google File System and of MapReduce. HDFS is a highly fault-tolerant distributed file system and like Hadoop designed to be deployed on low-cost hardware. Check out the following article for an introduction to Hadoop.