Every standard die is what's called a uniform distribution, meaning each face has exactly the same probability of landing face up. On a d6, you have a 1 in 6 chance (about 16.7%) of rolling any particular number. On a d20, each result has a 5% chance.
Things get more interesting when you roll multiple dice. Rolling 2d6, for example, doesn't give you an even distribution. Results near the middle (like 6, 7, or 8) come up far more often than the extremes (2 or 12), because there are more combinations of two dice that produce those middle values. That's a bell curve effect, and it's why games that use multiple dice tend to feel more consistent than ones using a single die.
On the technical side, online dice rollers use a pseudorandom number generator (PRNG) to produce results. Modern PRNGs are statistically robust enough for gaming purposes. The outputs pass standard randomness tests, so the distribution is fair. It's not truly random in a physics sense, but for rolling dice in a game, it absolutely holds up.
If you're curious about specific probabilities, like your odds of rolling a 20 on a d20 or getting at least a 15 on 3d6, probability calculators can break those numbers down. For most gaming situations, though, understanding the basics is plenty.