Bare with me on this one - this is very important. And there are videos further down if you scroll a bit.

Odds

Odds is the basic fundament of all gambling. If you do not have a very thorough understanding of odds when you are playing poker - then you are basically left with your luck. Your skill in other parts of the game will not take you that far.
The good news is that once you have really grasped how to use odds, it quite quickly becomes second nature.

Let me give you an example: A friend is proposing the following game to you. You will roll a six-sided dice and each time a 1 or 2 comes up you will win $2. Each time any other number comes up your friend wins $1.
It is quite easy to see that this game will even out in the long run, and neither of you will have made any gain from it. Each time you decide to roll, you can say that you are betting $1 to get a chance to win $2. Your friend is offering you 2:1 on your money if you win. That is called to have win odds of 2:1.
Your chance of winning each time you roll is 2/6 = 1/3 since over the long run, if you roll the dice 6 times it will show a 1 or a 2 two times.
Another way of putting that is that 1 out of 3 times you will loose - so your friend will win the other 2 out of the 3 times. You could say that your win odds are 2:1 or that the probability of a win is 33,3 %.

So how did I come up with that? There are two common forms of notation. The first one is the odds-against notation, which is often used in gambling (and also in poker). An example of this is 4:1 (read: four to one) or 2:1 (read: two to one). What this actually means is how often something won't happen compared to how often it will happen. 4:1 chance of winning actually means that you lose 4 times and win once. Note: it does not mean that you win 1 in 4 times, but 1 in 5 times. Converting odds into percentages is very simple. Divide 100 by the sum of the numbers left and right. So 4:1 is equal to 100/(4+1) = 20%. To change a percentage to odds you do the following: Odds = X:1, where X = (100/percentage) -1. So 25% is equal to 3:1 because X = (100/25) -1 = 4-1 = 3.

Now if your friend says the game stays the same but he believes he is so lucky that he is willing to pay you $2,5 each time you hit your numbers.
Then you would be getting odds of 2,5:1 and your chance of winning will still be the same. The money odds will be better than the winning odds which makes it a good bet for you.
In the long run you will win money from taking on this bet at those odds and your friend will loose.

Pot odds

Pot odds is the odds that you are getting from the money that is in the pot right now based on how much you have to pay to call.
For example, there is $10 in the pot and your opponent bets $5. Your pot odds would then be 3:1 based on that there will be $15 in the pot and you would risk $5 to get a chance of winning those $15.

Implied odds

Pot odds does not take into account that there might be further action in a hand. For example there might be opponents behind that could get involved or if you are on well concealed draw you might win a lot of extra money from an opponent with a strong hand.
Implied odds is your best guess on what your odds will be at the end of the hand.

Let's say for example that you are on a straight draw. The pot is $10, your opponent bets $5. You think that you will win an extra $10 in average if you hit your draw on the next card. No one else is involved in the hand and the next card is the last. Then you will pay $5 to get a chance of winning $25, i.e. your implied odds are 5:1. Especially in No-limit play, it is not uncommon that playing a hand based purely on pot odds would look like a bad bet, but when you take into consideration the extra money you will win (in average) it becomes a clear call (or raise).

Outs

An out is a card that will help you get a hand that you believe is stronger than your opponents. For example if you have an open ended straight draw (you have 4 consecutive cards) there are 8 cards that will help you make your straight - 4 on each end. These cards are called outs.
There is a rule of thumb for Texas Hold Em and Omaha that is accurate enough in most cases which says that each out give you 2 % chance of making your hand with one card to come and 4 % chance of making your hand with 2 cards to come.

The straight draw then in this example would give you 16 % chance of improving with one card to come.