Both MapReduce mode and local mode seem same to the user but the difference is the way they execute.
MapReduce mode:
In MapReduce mode, Pig script is executed on Hadoop cluster. The Pig scripts are converted into MapReduce jobs and then executed on Hadoop cluster (hdfs)
Local mode:
In this mode, Pig script runs on a Single machine without the need of Hadoop cluster or hdfs. Local mode is used for development purpose to see how the script would behave in an actual environment.