Solder paste is a material used in the manufacture of printed circuit boards to connect surface mount components to the copper traces of the board.  

The paste initially adheres components in place by being sticky, it is then heated (along with the rest of the board) melting the paste and forming a mechanical connection as well as an electrical connection.
  

The paste is applied to the board by silk screening and then the components are put in place by a pick-and-place machine.