The doc https://spark.apache.org/docs/1.1.0/submitting-applications.html
describes deploy-mode as :
--deploy-mode: Whether to deploy your driver on
No, when deploy-mode is client, the Driver Program is not necessarily the master node. You could run spark-submit on your laptop, and the Driver Program would run on your laptop.
On the contrary, when deploy-mode is cluster, then cluster manager (master node) is used to find a slave having enough available resources to execute the Driver Program. As a result, the Driver Program would run on one of the slave nodes. As its execution is delegated, you can not get the result from Driver Program, it must store its results in a file, database, etc.