I just want to know in simple language exactly, What the Syzygy tablebases are, Why they are used, What are their advantages, disadvantages?
Tablebases are databases of endgame positions, with few pieces left on the board (say a 5-man tablebase, which consists of all legal positions with 5 pieces on the board in total, including kings, say K+Q vs. K+R+B). These databases have the positions 'connected', i.e. they also contain the moves to get from one legal position to another (in particular, moves that eventually lead to mate). This allows a chess program, in such positions, to look straight into the tablebase, rather than to calculate via the engine.
Tablebases are perfect, in the sense that it can tell you number of moves to mate, and playing a position against a program using tablebases (in positions with very few pieces left) is akin to "playing with god". So as an advantage, they save computational time and allow your machine to play certain endgame positions perfectly. The disadvantage is that they take up a large amount of memory space.
Owing to the memory issue, compression techniques help to reduce the space taken up by the tablebases. Syzygy tablebases are relatively new (introduced about 2 years ago), compared to the Nalimov tablebases which have been around for quite a while (itself being a compression improvement over previous tablebases). The Nalimov 6-man tablebases take up over 1TB of memory space, whereas the Syzygy bases use less than 200GB.
The different names in front of 'tablebases', e.g. Syzygy and Nalimov, refer to nothing more than compression methods. For all practical purposes, there is absolutely no difference between the content of Syzygy and Nalimov tablebases; a computer that uses either one will play at the same strength (perfect!). Their only difference is the memory space they occupy.
(The possible difference in content, if any, is subtle: first off, every legal position in the tablebase has an evaluation attached to it (draw/win/loss in X moves). However, some positions may contain a sequence of moves to mate that require breaking the 50-move rule; the evaluations of these positions in the tablebases may be changed to accommodate for this, e.g. W/L/D/"win but break 50-move rule", but in any practical game, you are pretty much guaranteed that you will never see this difference.)