Dungeons & Dragons set the stage for using different sorts of RPG dice back in the day.
When we talk about dice, we use a convenient sort of shorthand. We use a lower-case 'd', followed by the number of sides the die has. So d6 is a regular six sided die, and d10 is a die with 10 sides. If you're new to role playing, you might be surprised to find out that there's also a d4, d8, d12, d20, d100, and more. Some games, like Fudge, even use custom designed dice!
Why the different numbers of sides on dice? Rolling a die represents randomness in the game. If you think that there's about a 50% chance of something happening, you can roll a regular d6 and say, "If it rolls 3 or less, then it happens". But if you think there's about a 5% chance of something happening, you can't do that on a d6. The closest you can get is about 16% (1 in 6). But on a d20, you can get exactly 5% by looking for a 1 to be rolled.
The sort of dice you need to buy depends on the game you're playing. We'll talk about a few games here, but check the rules of the game you're going to play to see what you'll need.
Dungeons & Dragons
Dungeons & Dragons uses a variety of dice. It's based on a d20 roll, but uses other sorts of dice for weapon damage, hit points, etc. At a minimum, you'd want a d4, d6, d8, d10, d12, and d20. But the chances are good that you'll need multiple of the d4, d6, and d8.
Call of Cthulhu
Call of Cthulhu is based on a d100 (percentile) system. This involves rolling two dice, both 10 sided. One die is marked with the numbers 0 through 9, and the other with the numbers 0, 10, 20, 30, 40, 50, 60, 70, 80, 90. You read the total result in terms of the sum of the two dice. So rolling a 10 and a 7 means your total result is 17.
You'll also need a few other dice for weapon damage. An extra d4, d6, d8, and d10 would be plenty, unless you get lucky enough to get hold of some serious weaponry.
Fudge is one of those games that uses it's own dice. Fudge dice have six sides, but they aren't numbered. Instead, two sides are blank, two sides have a plus sign, and two sides have a minus sign. You roll four of these together to get a relative modifier from -4 to +4 that applies to whatever skill you're using.
Luckily, you only need those four dice to play Fudge.
You can spend an enormous amount of money on dice. Fancy sets that include just a d4, d6, d8, d10, d12, d20, and d100 can run as high as $50 or more. But if you're just getting started, you probably don't want to spend that much on dice.
Buying plain mismatched dice can bring your cost down quite a bit. You can find the same set of dice, but worse looking and not all the same color, for under $5. Or you can buy individual dice to get exactly the set you want, and spend just as little.
I tend to use GM Dice online, but the Pound of Dice from Chessex is also attractive, if you don't care about getting factory seconds.
What to do when you don't have exactly the dice you need? Some substitution is possible without changing the probabilities.
For example, if you don't have a d100, you can roll two d10 and multiply the result of one of them by 10. You need to say which one is the 10s die before you roll, though! Or if you don't have d10s, you can use two d20s, and eliminate the left most digit. So a roll of 12 would be a 2, and a result of 20 is 0. As with the two d10s, multiply one of the d20s by 10 after eliminating the left most digit.
Let's say that you have a d20, and need to make a d12 roll. You can roll your d20, and if the result is over 12, just roll again and again until you get a result of 12 or under.
The simple act of asking to borrow someone's dice can often result in surprising results. Many gamers are very sensitive about their dice, feeling that they've formed a relationship with the dice over time. Having other people use their dice doesn't sit well with those folks.
On the other hand, many people don't care about sharing dice. So go ahead and ask if you need to, but keep in mind that you might get refused. Don't take it personally!
You can just carry your dice around in your pocket, but the dice bag is as much a part of role playing as the dice themselves. You can find any number of bags made for dice; these generally are a soft fabric, and have a drawstring closure.
You may even see people using Crown Royal bags as dice bags. If you like that look but don't drink, you can find these being sold online.
Regardless of the sort of bag you want to use, get one! Otherwise your dice will end up all over the floor eventually.
Other Sorts Of Dice
You can also find all sorts of dice out there. You can find three sided dice, dice marked for the game Rock Paper Scissors, thirty sided dice, and more. Hunt around if you're interested in odd dice, but be aware that you don't need those for most role playing games.
If you start buying those sorts of dice, the chances are good that you're hooked!