user13544: Below is the 1997 game of Deep Blue VS Garry Kasparov:
r1bqkb1r/pp1n1pp1/2p1pn1p/6N1/3P4/3B1N2/PPP2PPP/R1BQK2R w - - 0 1

The last move 7...h6? was played by Kasparov (black), after which Deep Blue played the knight sacrifice 8.Nxe6.
Wikipedia says:

The computer is aided by having this knight sacrifice programmed into its opening book. This move had been played in a number of previous high-level games, with white achieving a huge plus score. As an indication of how far computer chess has progressed in the 20 years after this match, modern programs deprived of their opening books are able to correctly evaluate Nxe6 as strongest; but at the time this was played it was considered probable based on other programs' performance that it was only the opening book that was responsible for this choice. The compensation white gets for the material is not obvious enough for the computer to see by itself.

How would white compensate for their sacrificed knight?

user12540: Honestly the Sacrifice of the Knight compensates with one pawn and a check of Bg6+ which forces the King to an unhealthy square of e7 . There is definitely an initiative which White has in the rest of the game but I will suggest that this move order is good for players like Mikhail Tal / Robert Spielmann / Rashid Nezhmetinov who has a daring style of play and sacrifice is their main weapon in major chess games . It is indeed difficult to answer and no fixed algorithm as how White will bring out the game and win it . Infact it could be Black who has better chances . 
True Black is cramped with King in Centre but Black has all its pieces surrounded and White has nothing except a Bishop on g6 which takes off some squares but with good play Black should be able to neutralize White's attack .After Qc7 or Qa5 Black's king will be safe on d8 with no immediate attack from White . 

user11609: History of this line is interesting. People used to think it's good for white, maybe majority still do as well as openings books do. But in correspondence database I use there are four games with Nxe6 fxe6 and black scored two wins and two draws out of four games. The wins are from 2012 and 2013. I know a lot of white players avoiding the sacrifice these days as it is easy to lose even against much weaker opposition while it doesn't even have the objective quality. Black play is easier and more obvious. I don't like White's position, I like Black here. It is too closed without direct threats for such a material investment. I know mainly the defensive plans for black here as it is in my repertoire, ok white try to play on e-file, disrupt you with Ne5 and f7 fork threat, may want to play c5 with B/Qa5+ sometimes and even b2-b4-b5 as black is undeveloped for quite some time. It is positional sacrifice and very interesting one. 

13544: <p>Below is the 1997 game of Deep Blue VS Garry Kasparov:</p>

<pre><code>r1bqkb1r/pp1n1pp1/2p1pn1p/6N1/3P4/3B1N2/PPP2PPP/R1BQK2R w - - 0 1
</code></pre>

<p>The last move <code>7...h6?</code> was played by Kasparov (black), after which Deep Blue played the knight sacrifice 8.Nxe6.</p>

<p>Wikipedia says:</p>

<blockquote>
  <p>The computer is aided by having this knight sacrifice programmed into its opening book. This move had been played in a number of previous high-level games, with white achieving a huge plus score. As an indication of how far computer chess has progressed in the 20 years after this match, modern programs deprived of their opening books are able to correctly evaluate Nxe6 as strongest; but at the time this was played it was considered probable based on other programs' performance that it was only the opening book that was responsible for this choice. The compensation white gets for the material is not obvious enough for the computer to see by itself.</p>
</blockquote>

<p>How would white compensate for their sacrificed knight?</p>
 12540: <p>Honestly the Sacrifice of the Knight compensates with one pawn and a check of Bg6+ which forces the King to an unhealthy square of e7 . There is definitely an initiative which White has in the rest of the game but I will suggest that this move order is good for players like Mikhail Tal / Robert Spielmann / Rashid Nezhmetinov who has a daring style of play and sacrifice is their main weapon in major chess games . It is indeed difficult to answer and no fixed algorithm as how White will bring out the game and win it . Infact it could be Black who has better chances . </p>

<p>True Black is cramped with King in Centre but Black has all its pieces surrounded and White has nothing except a Bishop on g6 which takes off some squares but with good play Black should be able to neutralize White's attack .After Qc7 or Qa5 Black's king will be safe on d8 with no immediate attack from White . </p>
 11609: <p>History of this line is interesting. People used to think it's good for white, maybe majority still do as well as openings books do. But in correspondence database I use there are four games with Nxe6 fxe6 and black scored two wins and two draws out of four games. The wins are from 2012 and 2013. I know a lot of white players avoiding the sacrifice these days as it is easy to lose even against much weaker opposition while it doesn't even have the objective quality. Black play is easier and more obvious. I don't like White's position, I like Black here. It is too closed without direct threats for such a material investment. I know mainly the defensive plans for black here as it is in my repertoire, ok white try to play on e-file, disrupt you with Ne5 and f7 fork threat, may want to play c5 with B/Qa5+ sometimes and even b2-b4-b5 as black is undeveloped for quite some time. It is positional sacrifice and very interesting one. </p>
