If a process to create a number is truly random then if you have the same inputs to it it will produce different outputs, this can be observed when you track the time between emissions of radioactive isotopes which has no relation to past conditions. The radiation will have a certain half-life which dictates average rate of decay but that is only an average, the numbers that make up that average are still random.
People often think that random strings of numbers have to have each number equally common for it to be random. This is not true, many random variables have a higher chance of being close to the mean rather than far from it but the process to make these numbers is still random.
If you give any finite sequence of numbers we cannot say for sure if it is random, the sequence 5,2,6,2,9,1,4 looks random but there's no reason why, if you continued the sequence, it could go like 5,2,6,2,9,1,4,5,2,6,2,9,1,4,5,2,6,2,9,1,4 and turn out to be no random.
What matters for a sequence of numbers being random is that we cannot predict the next number with any amount of information and the number does not depend on the past numbers the sequence. In the first sequence this appears to be the case but in the second sequence we can see that the numbers do depend on the number which came 7 steps before it.
In tests where humans think up random numbers they tend to not be random because humans like to jump around on the number line and not repeat the same number when they have just said it. This is not really random because if the human says 2 then the next number is less likely to be a 2 and so the next number depends on the one that comes before it. It doesn't depend on it in the sense that you can predict the human wont say 2 or that you can predict the next number will be a 7 but it does depend on the previous number in the sense that the previous number being a 2 changes the probability that the next number is a 2. I hope I phrased this well enough.
We have tests for randomness which look for patterns relating a number to the previous ones that came up. The digits of pi pass all these tests and appear to be random yet we can calculate the digits and predict precisely what the 567th digit will be so the sequence is not random. These tests are not perfect and can perceive a sequence like the digits of pi to be random when in fact they are not.