How do these hands relate to the starting hands the other people at the table have? Using the binomial distribution and the hypergeometric distribution, I came up with this table:
The first column is the number of people at the table who have that hand, assuming that there are 10 people at the table. The Pair column says that the probability of exactly two people having a pair is 9.6%. The Ace column says that about 86.7% of the time at least one ace is dealt to the table and about 50% (34.8+13.5+1.8) of the time two or more aces are dealt to the table. So having an ace 2 for a starting hand means that it is about even odds that you do not have the best hand at the table with an ace in it.
The most useful part of the wikipedia page is the section detailing the approximation of hitting outs. Say that you have two spades and the flop gives you two more spades. That leaves 9 more spades in the deck or 9 outs to get a flush. Simply times the number of outs by 4 to get the odds of getting a flush by the river. 4*9 = 36 so 36% chance. This is an approximation with the actual probability being 34.96%. For 10 or more outs after the flop the formula is 3x+9 with x being the number of outs. The approximation on the turn is 2x (or the better approximation of 2x+(2x/10)). Pretty useful and gives you a handy way of calculating pot odds during a hand.
Before the flop what are the odds of improving the starting hands that I mentioned above? For a pair there is roughly a 11.5% chance of getting three of a kind on the flop, 15% by the turn, and 18.5% by the turn. Chance of four of a kind is .24%, .49%, and .82% respectively. If you have two cards of the same suit it is about a .8% chance of getting a flush on the flop, 2.8% of getting it by the turn, and 5.8% of getting it by the river. The odds of getting a straight or pairing one of your cards can be found using the post-flop approximations.
