Expected value is the average amount you can anticipate to gain or lose by acting a certain way.  It is the weighted average of all possible outcomes for a certain event.  A good poker player will use expected value when analyzing a bet, hand, hourly rate etc.  If you make your decisions based on positive expected values, then on average you will win more than lose.

An easy example, coin flips. Your expected value is zero because over a long enough period of time, odds state that the coin will land on hands just as many times as it will land on tails. Now if someone came to you and said they would give you $10 everytime the coin landed on heads, but you would have to give them $1 if it landed on tails, should you take the bet? Of course! You can expect to win $5 50% of the time ($10 x .5) and expect to lose only $0.50 50% of the time ($1 x .5), resulting in an expected value of $4.50.  

This is a simplified example, but is directly correlated to poker.  You can calculate your expected value for a poker hand, game, etc. the same way, the only difference is that the numbers usually won't be as simple to calculate.

Basically, expected value can be expressed in the following way:

EV = (p * w) - b

where

p is your probability of winning
w is the amount you will receive for winning
b is how much you are wagering or betting

Take the above example, your probability of winning is 50%, multiplied by the amount you could win of $10, minus how much you bet or could lose, $0.50.

EV = (50% * $10) - $0.50 = $4.50

The ratio w:b is an important ratio, it is your pot odds percentage .  The odds of winning the hand must be greater than your pot odds for it to be profitable to play.    

Pot Odds = w:b = $10:$0.50 = 20%

Learn more about pot odds in poker.