Theory
Fairness of dice is a sometimes loose concept. While there are games in which each result matters significantly (i.e. Yahtzee), there are games like pen and paper RPGs where there's a difficulty rating, and whether you succeed by rolling a 16 or 17 isn't important. Of course, a different check might succeed on a 17, but fail on a 16, but what I'm trying to get at is that not in all situations, perfect fairness matters. But of course, a perfectly fair die is never worse than a biased one in this regard.
Another thing I want to note is that while the question is about the die as an object itself, how it's being tossed is another matter entirely, which influences how fairly a die will actually roll, regardless of its physical capabilities of doing so, and even if you're not actively trying to cheat on the toss.
Manufacturing
As far as my research goes, accuracy is mainly attempted to be achieved at the production stage, so starting out with well-manufactured dice seems to be mandatory. This consists of getting a mold free of air bubbles, avoid bias from finishing the die (such as from tumbling). and mitigate the uneven amount of material taken out of the die when carving numbers into them.
Six-Sided Dice
If you're looking for six-sided dice, there's two great commonly available options: Casino dice, and precision Backgammon dice. Both are made with accuracy in mind, and should offer the highest amount of fairness you can wish to acquire due to the stakes involved in using them (referring to casino dice in particular).
Non-Six-Sided Dice
If you're looking for less common dice, such as the typical d4, d6, d8, d10, d12, d20, d100 assortment of pen&paper dice, the ones I've found to be well-manufactured are GameScience/Zocchi precision dice. There's this test that basically concludes that non-precision Chessex dice are good enough, however also shows that precision dice have a more even spread, which is what this question is asking for. There's a video by Louis Zocchi, explaining the reasons for that, basically stating that tumbling destroys the balance of the die.
I personally own four sets of these dice, inked them with crayons (which should mitigate the missing material more than a pen would, but I cannot provide any measurements in terms of how well this works, or even whether I potentially made it worse in terms of balance).
I believe I've come across precision dice from another company over the years, but never owned a set of them, and I failed to find them when looking for it now.
Opaque vs. Transparent Dice
Air bubbles may form within a die during the process of injection molding, which may result in an unbalanced die. That can be avoided (in part or full) by injecting the material more slowly, as well as letting the dice cure (cool) for a longer amount of time, which is done for transparent dice, where air bubbles would otherwise be visible. For opaque dice, this slower process isn't necessary to achieve a perfect appearance, as you cannot spot any air bubbles forming under the surface, so it makes sense for manufacturing efficiency reasons to speed up the process, resulting in the aforementioned air bubbles. This has apparently been confirmed by dice manufacturers, and makes sense if you consider that outside of this question thread, "good enough" works for most players. The previous link states that this might not actually affect the balance of the dice, which of course is true (especially considering that dice manufacturers may use different processes), but tests exist that at least some opaque dice are far more biased than transparent dice (although I will point out that this is certainly not a Chessex-specific issue - it's just what the person in the video happened to test).
Shape
Persi Diaconis introduced me to the concept of fairness by symmetry through his YouTube videos, going back to Euclid and Archimedes. The idea is basically that a perfectly symmetric die (which can be a d4, d6, d8, d12 or d20) is more fair than a non-perfectly-symmetric one, even if the latter features sides of equal shapes and dimensions. If I paraphrase him correctly, that is because of the ability to manipulate a non-perfectly-symmetric die more easily through use of manual dexterity, as well as a higher difficulty of predicting the "random" outcome for symmetric dice, even though the randomness in a die toss is technically only determined by physics and therefore in theory computeable. The concept is perhaps more of a philosophical than practical one, as your definition of fairness may differ from his, but I think it's worth mentioning in any case.
Evaluation
Whether you want to check if your precision dice are as accurate as they claim to be, or you want to test whether a set of non-precision dice is accurate enough, having a way to determine it for any particular dice is handy. There are three methods I know of to evaluate dice:
Rolling them a large number of times, and see how evenly spread the results are
This method is likely to give you a very good idea of the fairness of your dice, as long as you are willing to roll a ridiculous amount of times, and go through the effort to document every roll. Of course, keep in mind that this still is subject to probability, so you're not very likely to actually get even results, even for a perfectly balanced die. The test I already linked above should give a good idea of how a "good" result will look in terms of variance. In addition, Persi Diaconis states that "The notion of long-term frequency [is] actually pretty fictional" (source).
The Chi-squared test
The Chi-squared test is a more sophisticated method of analyzing dice rolls. Instead of leaving the interpretation of the results to you, it provides a formula to calculate whether your die's results have equal frequencies or not, based on the null hypothesis. It involves slightly more work as the previous one, as in addition to a sufficient amount of samples, you need to perform calculations with those numbers. However there are programs you can use to trivialize this task, although I dare not pick a favorite to link to at this point in time.
Spinning a die in salted water
This method takes the roll out of the equation and basically tests the balance of the die without any need for comparing results, making it a quick alternative to the above method. If your die appears to be spinning randomly and keeps showing different results, there's a very good chance that it's rather fair. I keep phrasing this rather ambiguously, because if your die has an even distribution between the majority of the numbers, but will roll a few of them more or less often than it should, there's a good chance you'll miss it using this method.
