Poker odds are an interesting concept and one that could be discussed at length for pages and pages at a time. When you consider the fact that whole books have been written by people like David Sklansky on the topic of poker odds, you will realize that it is very difficult to make you an expert at poker odds in the space of one short article. However, what one short article can do is give you a start on your understanding of poker odds and therefore allow you to apply what you learn here into whatever situation it is that you might have.
The reason that poker odds are both interesting and useful is because of the fact that you can use your knowledge of odds to determine whether a particular proposition is going to be positive for you over time. This is what people that invest in stocks and bonds do; they calculate risk and reward and if the ratio of reward to risk is equivalent to the chances of their stock doing what they think it will do or better, then they execute the trade. In poker it is the exact same thing and the two parts are the pot odds and implied odds (risk to reward ratio) and reading skills and understanding of poker odds (chances of the stock doing what you want it to do).
Pot odds, quite simply put, are the odds of the current pot size to the bet that you need to call in order to see the next card. Therefore, if the pot is $100 and you need to put $10 into the pot to see the next card, your pot odds are 10:1. Pretty simple stuff. Implied odds are the same thing as pot odds, except they take into account future streets by you considering what it is that your opponent might do. For example, if you have a pot of $100 and are facing a bet of $20 on the turn, your pot odds are 5:1.
However, if you think that your opponent is going to bet on the river regardless of what comes and you plan only to call if you make your hand, your implied odds are $120 ($100 in the pot plus the $20 river bet) to $20 (your $20 turn call) or 6:1. If, however, you were going to call on the river even if you did not make your hand (maybe you had a hand already that was worth a value call), then your implied odds are $120 to $40, or 3:1.
When talking about different poker variants, it is important to understand that the rules are going to be different. In board games, you can calculate your odds by looking at all of the cards you don't know (i.e. all your opponent down cards and the burn cards) and comparing that to the number of cards to come that can help you make your hand. The same reasoning applies in seven card stud and draw poker as well, keeping in mind that you must consider all of the cards you can not see in the denominator.
Therefore, the chance of you making your hand at any given point is the number of cards that can make your hand (your outs) divided by the total number of unknown cards. This principle is universal and while the application is different for different games, if you understand this principle you will be able to apply it to any poker game.