PNR stands for Passenger Name Record and is the unique 10 digit reference number which is issued and stored in Indian Railways database when you book a ticket from Indian Railways Reservation counter or IRCTC's Website. The PNR when issued hold all the information about the journey like Passenger or group of passenger name's limited to 6 and 4 in case of Tatkal Quota booking, status of the berth/seat alloted to each passenger, age, gender, boarding station, destination station, date of journey, quota and fare amount.

As mentioned above the PNR contains the status of the seat/berth alloted to each passenger traveling on a single PNR, sometimes when we book a ticket RAC (Reservation Against cancellation) or W/L (Waiting List) is issued instead of a confirmed ticket, in that case we need to check the current status of the PNR to find out if our ticket is confirmed or not.
You can check your PNR Status anytime, but its best you check your PNR Status on the day of your journey as on you journey date the Final Charting for the train you are traveling is done and all the last minute cancellations are done and quotas are adjusted. So its better you check your PNR Status on the day you are traveling.
Generally PNR number is printed on the right hand top corner of your Ticket, if you booked it through Indian Railways Reservation counter and on the left hand top corner if you booked your ticket from IRCTC Website. Additionally you can also find your PNR Number on the email and SMS if you booked your Ticket online.
PNR Prediction is process by which one can tell if your seat/berth will confirm or not. On RunningStatus.IN PNR Prediction is based on the past records, date of journey, number of passengers and other factors which the RunningStatus.IN algorithm takes into consideration and searches through the millions of records saved in RunningStatus.IN databsed and gives the prediction for each passenger traveling on single PNR.