One word for "The amount paid to a person A because of the damage caused to him by person B"

Solution 1:

In a modern civil law context, damages:

A sum of money claimed or awarded in compensation for a loss or an injury:

He or she sues not only for personal injury but for damages for the loss or destruction of the motor vehicle.

Also sometimes compensation:

Something, typically money, awarded to someone in recognition of loss, suffering, or injury:

he is seeking compensation for injuries suffered at work

Or restitution:

Recompense for injury or loss:

Since such legislation is not compatible, in my view, with Community law, they should, in principle, be entitled to seek restitution for those payments.

Occasionally recompense, sometimes as a noun but more often as a verb:

Make amends to (someone) for loss or harm suffered; compensate:

offenders should recompense their victims

There's also 'making amends' - that's usually used to mean non-financial compensation, although money could be involved.

In the context of war, reparations:

The compensation for war damage paid by a defeated state:

the Treaty of Versailles imposed heavy reparations and restrictions on Germany

Where state A has conquered state B in the past (or has shown that they are capable of doing so) and state B pays state A 'protection money' not to invade again, that money is called 'danegelt':

1911 C. R. L. Fletcher & R. Kipling School Hist. Eng. ii. 39

It is always a temptation to an armed and agile nation,
To call upon a neighbour and to say:—
‘We invaded you last night—
we are quite prepared to fight,
Unless you pay us cash to go away’.
And that is called asking for Dane-geld.

It's a pejorative term. Where a losing state is forced to pay money to a winning state, whether this is 'war reparations' or 'danegelt' might depend on your point of view, on whether you belive the amount is proportionate to the actual loses, on which state you believe was the aggressor, etc etc.

Ethelread the Unready. A Weak King

Rather than wait for him the Danes used to fine him large sums called Danegeld, for not being ready. But though they were always ready, the Danes had very bad memories and often used to forget that they had been paid the Danegeld and come back for it almost before they had sailed away. By that time Ethelread was always unready again.

(parody)

1066 and All That, Sellar & Yeatman, 1931.

Solution 2:

In the context of war the word is usually

reparation

It can also be used in contexts other than war ( http://www.merriam-webster.com/dictionary/reparation )