What is PLAR and how is it different from PLAR-OPX?
Is this the same as Hotdial or Autodial?
Private Line Automatic Ringdown (PLAR) is traditionally an analog telephony concept that implies the following:
- Private Line - No telephone company switches are involved. When a device is taken off-hook, it applies a ringing voltage to the circuit. Other devices on the same pair will ring. Then, when another device is answered, a call will be maintained over the circuit at normal voltage. The telephone company switch is not involved in the process as there is a fixed destination defined, making this a private line. 
- Automatic Ringdown - A two-point automatic ringdown circuit, or ringdown, has a telephone at each end. When the telephone at one end goes off-hook, the phone (or phones) at the other end instantly rings. There is no dialing required and handsets without dialing capability are often used to avoid confusion. Many ringdown circuits work in both directions. In some cases a circuit is designed to work in one direction only. That is, going off-hook at one end (end A) rings the other (end B). Going off-hook at end B has no effect at end A. 
This type of phone is very frequently called a "batphone" - in reference to the red phone only used to reach Batman. I've never personally used PLAR to configure a phone to reach a superhero. Luckily, there are many other possible uses for PLAR. A hotel concierge may use a phone that directly connects to a taxi company, for example. Another usage of PLAR is when the remote end needs a reasonable expectation that the caller is known and trusted, for example, a handset in an operating room that can only reach an OR nurses station.
If you are preparing for any of the Cisco voice certification exams, be sure that you are familiar with the "connection plar" IOS command used on Cisco gateways to provide the PLAR functionality.
The terms "Hotdial" or "Autodial" are also used frequently to describe the same functionality as "PLAR". In my experience, I've seen the term "PLAR" used more frequently when a router/gateway is used to provide the automatic ringdown and the term "Hotdial" to be used in CUCM environments. But, with anything, your experience may vary.
Private Line Automatic Ringdown - Off Premises Extension (PLAR-OPX)
The OPX mode allows remote users on FXS ports to appear to a central PBX as a directly connected extension. When the FXO port detects a ring signal from the PBX, the router sends a VoIP call setup to the remote FXS port but it does not take the FXO port off-hook. As a result, the PBX only sees the call answer signal when the RemoteSide router FXS port is picked up. After the PBX reaches the no answer timeout (call rings out), then it can end the call, transfer the call to voice mail, or ring another extension/ring group. Without OPX mode, the FXO port immediately goes off-hook after it senses the ringing and the PBX is then unable to perform a call forward, no answer, or roll over to voice mail. 
When reviewing a gateway configuration, you will know this to be the case when the "connection plar-opx" command is observed.
PLAR, Hotdial & Autodial Sample Configurations
Sample Configurations can be found here:
- Gateway Configuration
- Cisco Unified Communication Manager Configuration
- Hookflash Relay Configuration in a PLAR-OPX Scenario