Privilege Escalation is the process where the attacker gains un-authorized access rights or higher-level privileges of software/program, network or system. This is the next phase of a cyber security attack after the Footprinting, Scanning, Enumeration, Vulnerability Analysis and Exploitation.
Horizontal Privilege Escalation is where the attacker tries to gain the access other users like file shares, network access, applicatioin access etc.
Vertical Privilege Escalation is where the attacker tries to gain higher privileges belonging to an existing user like getting root access of a server.