If you are at all familiar with RPGs, you will probably know most of this stuff already, but read it anyway, as we probably do things slightly differently than you expect.
The dice you will need for this game are a six-sided die, and two ten-sided dice (preferably different colours). One such set for everyone playing will make everything go much faster.
The ten sided die is referred to as a "d10" and is numbered from 0 to 9. Unlike many other role-playing games where a roll of "0" on a d10 counts as a ten, in Infinite Horizons, it is zero (if your d10 goes from 1 to 10, the ten should be read as a 0).
The six-sided die, just like the one used in almost every board game since the beginning of time, is called a "d6".
Another common requirement is the percentile die roll, denoted as "d%". You can get a hundred-sided die for this, but these dice are expensive, and not usually as random as rolling two d10s, taking one as the tens digit, and the other as the ones digit. This is the reason for having two different colour d10s; you can always say "the purple one is the tens" before rolling, and then not have to worry about whether that 1 and 9 is a 19 or a 91 (or be rightly accused of cheating if you always pick the more advantageous of the two). In Infinite Horizons, a roll of "00" on a d% is read as a 0, not as 100 as in many games; the possible range for a d% roll is 0 to 99.
If you see "3d6", this means you need to roll three six-sided dice (or roll one six-sided die three times) and add up the rolls. Similarly, "5d10" means roll five ten-sided dice, and add the rolls together. Remember that the range for a d10 is 0 to 9, so the range for 5d10 is 0 to 45, not 5 to 50.
If you see "d6+3", this means roll a d6, and add three to the result. "2d6+8" means roll 2d6 and add 8 to the total; do not roll d6+8 twice and add the two numbers! "d10-4" means subtract four from the number rolled on a d10. No die roll can go below zero; any negative result is treated as zero (if you roll anything from 0 to 4 on a d10-4, read the result as 0).
There are many dice made with numbers of sides other that 6 and 10 (we have seen 4, 7, 8, 12, 20, 30, 100, as well as a d6 marked to be a d2 or d3, and there are almost certainly others). These are not necessary, although they may be helpful sometimes. For example, if you need to randomly determine the day of the month something happens on, a d30 would be the easiest way (assuming 30 days in the month). However, by using a d6 (treating 1 or 2 as 0, 3 or 4 as 1, 5 or 6 as 2) as the tens digit, and a d10 for the ones digit, the same result may be achieved. Or, you could just roll d% until you get a number between 1 and 30. If something was going to happen between 5:00 and midnight, you could use d8+4 for the hour, or you could use a d10+5, rerolling any 13 or 14. In short, any roll made by any of these custom dice may be emulated easily with your d6, d10, d%, and a little imagination.