Pure Strategy:

By pure strategy we mean a player is going to play her strategy for sure (that is with probability 1). There is no randomness associated with this strategy. 

Mixed Strategy:

Whereas  by  mixed  strategy we  mean  a  player  is  playing  her  strategy with some probability  attached  to  it.  Suppose, for example,  we  say  if  a  head appears  in  a coin  toss, then we  are going  to the movie, but  if  a  tail  appears, then  we  are  going to the museum. Here our strategy  is random  as  there  is probability attached to our strategy choice.  

Payoffs:

In a game, each  player is given a complete numerical scale with which  to compare all logically conceivable outcomes of a game, corresponding to each available combination of choices of strategies by  all the players. The number associated with each possible outcome will be called the players' payoff for that outcome. Higher payoff number attach to outcomes that are better  in  this player's rating system. Sometimes the payoff will be  simple numerical ratings of  the  outcomes. In games where the strategies are continuous (for example, output decisions of a firm in Cournot game and Stackelberg game, bids to acquire a tender, etc.) the payoffs of the players could  be  generated by  a  function where the arguments of the function are the strategies chosen by  the players.