Why Relay often has two ratings AC and DC...? How to choose the correct one ...???

The contacts in a relay (and the same is true for other switches) have to be able to safely and repeatedly allow and interrupt the rated current. 

Interrupting is hard on the contacts, since there is always some inductance in the circuit, which creates a high voltage when the current is interrupted.  This voltage causes a spark (continued current flow) when the contacts open, and gradual damage to the contact surfaces. If the contacts are too close, and the circuit can supply a high enough voltage, the spark may continue a long time.

In DC circuits the spark is only extinguished when the contacts are far enough apart. The higher the voltage of the circuit, the bigger the contact gap has to be.  In AC circuits this spark stops naturally because voltage and current reduce to zero twice per AC cycle, so for a given contact separation a much higher circuit voltage can be tolerated.

How to calculate for the Correct Relay

Relay Ratings and Limits

Relays often have two ratings: AC and DC.  These rating indicate how much power can be switched through the relays. This does not necessarily tell you what the limits of the relay are. For instance, a 5 Amp relay rated at 250 VAC can also switch 10 Amps at 125VAC. Similarly, a 5 Amp relay rated at 48 VDC can switch 10 Amps at 24VDC, or even 20 Amps at 12VDC. 

Volts x Amps = Watts - Never Exceed Watts!

An easy way to determine the limit of a relay is to multiply the rated Volts times the rated Amps. This will give you the total watts a relay can switch.  Every relay will have two ratings: AC and DC.  You should determine the AC watts and the DC watts, and never exceed these ratings.  

Example Calculations:

AC Volts x AC Amps = AC Watts
DC Volts x DC Amps = DC Watts

Example:
A 5 Amp Relay is Rated at 250 Volts AC.
5 x 250 = 1,250 AC Watts

Example:
A 5 Amp Relay is Rated at 24 Volts DC.
5 x 24 = 120 DC Watts

If you are switching AC Devices, Make Sure the AC Watts of the Device you are Switching DOES NOT Exceed 1,250 when using a 5A Relay.
If you are switching DC Devices, Make Sure the DC Watts of the Device you are Switching DOES NOT Exceed 120 when using a 5A Relay.

Resistive and Inductive Loads

Relays are often rated for switching resistive loads.  Inductive loads can be very hard on the contacts of a relay.  A resistive load is a device that stays electrically quiet when powered up, such as an incandescent light bulb.  An inductive load typically has a violent startup voltage or amperage requirement, such as a motor or a transformer.' 

Startup and Runtime Loads

Inductive loads typically require 2-3 times the runtime voltage or amperage when power is first applied to the device.  For instance, a motor rate at 5 Amps, 125 VAC will often require 10-15 amps just to get the shaft of the motor in motion.  Once in motion, the the motor may consume no more than 5 amps.  When driving these types of loads, choose a relay that exceeds the initial requirement of the motor.  In this case, a 20-30 Amp relay should be used for best relay life. 

When the relay opens it will draw an arc. With AC power the current drops to zero 100 or 120 times per second (depending on whether you have 50Hz or 60Hz power), and this will allow the arc to extinguish. With DC power you don't get this automatic interruption, and an opening arc may last longer, burning the contacts in the process. That's why relays are allowed to switch only a fraction of the AC power if DC.
It's not uncommon to see 250V AC relays only rated for 30V DC.