shaytaan: The rebellious haughty one; the deviated one (who) deviates the slaves of Allaah; the one who hinders them from obedience to Allaah (The Blessed and Most High).
ar-rajeem: The banished one; the cursed distanced one, whom Allaah distanced from His Mercy. And when it is the case that he is distanced from Allaah’s Mercy, he wants to distance the slaves of Allaah from Allaah’s Mercy. Therefore, a slave (of Allaah) is required to seek Allaah’s Refuge against this shaytaan -the banished haughty one who works to divert the human being from obedience to Allaah, the worship of Allaah and (from being) successful through Allaah’s Mercy.
[Source: Sharh Ad-Duroos Al-Muhimmah Li-Aammatil Ummah’ by Shaikh Abdur-Razzaaq Al-Badr (may Allaah preserve him).. page13. slightly paraphrased]