What are "pick-up", "dropout" and "rated" voltages in a Relay datasheet?

Rated voltage is the coil voltage that particular model is designed for. (The first page of the datasheet shows that the last digit of the model number is the coil voltage.)

As for the others, the "pickup voltage" shows that the relay is guaranteed to not require more than 70% of the rated voltage to switch the contacts "on". It may turn on with less voltage but this is not guaranteed.

"Dropout voltage" shows how low the coil voltage must go to turn "off" again. It will always turn off at 10% or lower. The value may be higher (though not as high as the rated pickup voltage.

So, to put it another way, if you are using a 5 volt model, and you make sure that the on voltage fed to it is at least 3.5 volts (70% of 5v) and the off voltage is no higher than 0.5 volts (10%), the relay will work as you would expect. If the on voltage is lower or the off voltage is higher, it may or may not switch.

Pickup voltage is the minimum voltage at which the relay is guaranteed to pull in (similar to Vih for a digital gate).

For a later reader, to clear out the meaning of min and max in table "70% max and 10%min", as it was demanded, the terms are really confusing. The "max" means that the voltage may be less then 70% to pull in, but 70% is the maximum value of all voltages applied starting from where the manufacturer guarantee to pull in. Or in other words, the maximum voltage that is not guaranteed to pull in, and after that (higher) is guaranteed

Dropout voltage is the maximum voltage at which the relay is guaranteed to drop out after it has been pulled in (similar to Vil for a digital gate). The "min" means that the voltage may be greater then 10% to drop out, but 10% is the minimum value of all voltages applied starting from where the manufacturer guarantee to drop out. Or in other words, the minimum voltage that is not guaranteed to drop out, and after that (lower) is guaranteed. Hope that help understanding.

Relays generally have a lot of hysteresis, meaning that once the relay is pulled in it takes much less current to keep it pulled in (unless you whack it and open the magnetic circuit).

You should be aware of a bit of subtlety here that other answers are glossing over.

Relays are current-operated devices- and generally the coil is a winding of magnet wire. That means that the little note (2) on the datasheet (like many such 'fine print' notes) is very important, particularly if you wish your design to reliably operate over a range of conditions. The specs are in terms of applied voltage but the relay only really cares about current (because the mechanical spring constant and magnetic characteristics do not change much with temperature and because of Ampere's law).

Copper increases in resistivity with temperature (by about +0.4%/°C).

The relay is guaranteed to pull in when a voltage of 70% of the rated voltage is applied at 23°C coil temperature. The coil can get hot from the environment and it can get much hotter as a result of the current flowing through it. There is often a separate spec for the 'hot start' condition. If the coil temperature is 100°C and the initial resistance was 720 ohms @ 23°C it will now be 936 ohms, and the current will be reduced to 77% of its value at 23°C. Suddenly that margin does not look so great. A 10% reduction in voltage means the relay may not pull in at all.

Extended temperature relays (with special high temperature insulation rating such as 'H' 180°C rated) may not be guaranteed to pull in at all even with the full nominal voltage applied.

The same effect exists with the drop-out (the minimum voltage is reduced at very low temperatures) however it is less of an issue in most cases because we can usually reduce the coil voltage to almost zero, especially at low temperatures where devices leak less. Your 720 ohm coil would be 543 ohms at -40°C so you need to keep the coil voltage under 900mV (not 1.2V) to ensure drop-out.

As you might expect, this must be considered in applications such as automotive.

As well, coil suppression (eg. flyback diode) or low supply voltage will make the relay switch significantly more slowly and thus will reduce the contact life. The specified life is generally without those factors included.

TL;DR: Drive the relay coils at the nominal voltage in most cases.

Rated (coil) voltage
This is the nominal voltage the coil has been designed to work with. Applying a (much) higher voltage would be a waste of power and could potentially destroy the relay due to heat or mechanical stress.

Pick-up voltage
The pick-up voltage relates to the minimum initial force required to overcome the static friction to switch the relay on. If you apply at least 70% of the rated voltage, the relay is "guaranteed" to switch on. However, one would normally use the rated voltage to have some security margin.

Dropout voltage
The dropout voltage is the voltage that is required to keep the relay switched on. When the voltage falls below 10% of the rated voltage, the relay switches off.