(Note: Due to this FAQ not being one that scientific experiments can directly answer, the following is the opinion of the editor 'Silverhydra'. Information should be taken with some skepticism.)
Firstly, supplements are not needed to be 'healthy'. Health is a vague enough concept as it is, and there is the old adage that 'supplements are there to supplement your diet'.
If you want to take supplements for health and you are determined to take some manner of supplement, then proceed to the third header entitled 'Micronutrient and Macronutrient Supplementation'. It is not our position to question your habits or force a change to your lifestyle, changes to your lifestyle should be made autonomously.
If you want to get 'healthy' regardless of what that entails, continue on to the next header.
In regards to diet, no specific diet will be put forth in this section as an ultimate one for 'health'.
The general idea that you want out of a diet is:
It is sustainable for a long period of time (unless otherwise intended)
It does not promote weight gain unless needed (ie. underweight) and is substantial enough to prevent unwanted weight loss (intended weight loss is fine). Basically, you have a rough grasp on caloric intake.
It includes a variety of foods. Some food groups may be omitted, but caution must be taken for the nutrients that are now lacking in your diet.
You actually enjoy it. Diets are lifetime things, it would suck to hate the rest of your life when it comes to food.
In regards to exercise, there are benefits to both cardiovascular/aerobic work and there are benefits to resistance training (weight lifting). In a perfect world, one would indulge in both.
If you do not want to, or do not have the time to indulge in both, then do what you prefer. Some activities are more geared towards certain goals (ie. weight lifting to build muscle mass) but most health benefits come from merely having some form of moderately or highly strenuous activity.
As for habits, some habits are adverse to health when excessive. These may include:
Smoking cigarettes (although smoke itself can be dangerous, and thus would extend to marijuana unless vaporized or brownie-ized)
Excessive alcohol consumption or binge drinking
Otherwise unmentioned substance abuse (including some supplements, such as stimulants. Keep in mind that supplements are 'substances')
A lifestyle conducive to stress
Try to minimize unhealthy habits.
This is the first 'tier' of supplements for health. And for the most part can be wholly avoided if your diet is good enough.
One must strive to have all 24 essential vitamin or minerals for health, most of them at or around the RDA (the 100% on food labels). The 100% value is actually based off a bell curve (normal distribution) and some people require more or less that value. Don't be too strict on getting to this number, just try to get them all in adequate amounts.
You can also overconsume on one day if you underconsumed the day prior; the body tends to store these nutrients, and adverse health effects occur if you fail to ingest a nutrient significantly and for a long period of time.
Please note that out of the 24 vitamins and minerals, only a few of them are commonly deficient. Magnesium may be a concern due to deficiencies being relatively prevalent, and Vitamin D has had its requirements bumped up from the RDA of 400 IU to around 1,000 IU to 2,000 IU daily. Additionally, athletes may be deficient in Zinc if they sweat excessively. Women at risk of osteoporosis may benefit from a Calcium-D-Glucarate supplement as well (although this can be avoided if you use a Protein Supplement that is derived from milk, such as Whey Protein or Casein Protein, due to their high calcium content).
Despite the prevalence of iron-deficiency anemia, it is best to counteract that through dietary means (such as an increase in animal iron sources, pairing of non-heme iron with vitamin C, or cooking in an iron pot with an acidic medium) due to excessive iron being a potent pro-oxidant. This is overridden if prescribed iron by a medical professional, however.
Be aware that if you buy a multivitamin supplement, that some nutrients may not be given in adequate dosages. Calcium and Magnesium are notoriously underdosed in most multivitamin supplements.
As for macronutrients, if your diet is lacking in protein and getting additional protein through dietary means is either unpalatable or infeasible (perhaps expensive), consider buying a Protein Supplement to supplement your protein intake to what your goal may be.
If you do not intake fatty fish routinely, then a Fish Oil supplement may be considered.
Other supplements, in this category, are supplements that tend to act like vitamins in the body. These supplements are found in food, but have been otherwise shown to be beneficial in quantities beyond what is found in food (and thus, supplements).
In regards to the aforementioned 'this article is written by an editor and thus should be taken with a bit of skepticism', this is the part where it applies most.
Supplements that would support health, and should be considered, are:
Creatine, typically in the form of monohydrate
CoQ10, also known as CoEnzyme Q10
Many other supplements can also give benefits to health, but the above four seem to be very well studied and safe to consume.
Additionally, I (Silverhydra) do not wish to put forward too many at once. If you frequent Examine you will come to learn of new supplements and new science, and may stumble upon new things to try eventually (like those sexy Bioflavonoids). For now, consider the above four and 'calibrating' the diet the basics of supplementation.