Vitamin C is perhaps the most common dietary supplement used; it is used for a variety of purposes including cold symptom reduction, cancer treatment, and for a plethora of reasons related to its antioxidant properties.
In regards to cold symptoms, which appears to be the most popular usage of vitamin C, it appears to be somewhat effective at reducing the duration of sickness when taken at supplemental dosages (1,500-2,000mg) in youth or adults, either at the onset of sickness or as a daily preventative. The actual frequency of sickness does not appear to be altered in normal adults or children, but may be halved in those subject to strenous exercise such as marathons (these persons tend to have drastic increases in risk anyways, so vitamin C may be normalizing the increased risk). Vitamin C is not a cure for the common cold, but it may have a role regardless.
Vitamin C does confer a variety of health properties, which can probably be divided into whether vitamin C is preventing a deficiency (intakes of 100-200mg or so) or being used in pharmacologically high doses (1,500-2,000mg, or more). The health properties associated with the former are most likely irrelevant when considering the benefits of supplemental vitamin C since it is already very prominent in the human diet, but normalizing a deficiency (and not getting scurvy) reduces fatigue and improves cognition alongside benefits to the body's mucus membranes (most visibly, your gums don't bleed at random).
Pharmacological (supplemental) dosages of vitamin C are implicated in elevating mood according to one study (under the depression section of neurology) and as mentioned before, reducing cold frequency in hard training athletes. There is also a bidirectional influence on cortisol seen with vitamin C, as it can suppress the increase in cortisol seen during exercise if the exercise is kept to a minimal degree but as soon as the exercise is as strenuous as a marathon or day-long skiing session vitamin C actually augments cortisol secretion. The aforementioned properties are somewhat unique to vitamin C.
Other health effects include promoting blood circulation in persons with impaired blood circulation (not necessarily in healthy controls), neuroprotection secondary to NMDA antagonism, and protection from smoking or prooxidant toxins. These properties, however, are solely due to the antioxidant properties of vitamin C and can likely be mimicked by other antioxidants (the benefits to blood circulation seen with vitamin C, for example, are significantly outperformed by pycnogenol via the same mechanism).
In regards to intravenous vitamin C, while it may have some promising usages due to the inherent prooxidant effects of excess vitamin C it is currently not recommended unless under the supervision of a medical doctor due to the risk of renal oxalate nephropathy (a treatable condition is treated quickly, but can potentially be lethal if left untreated).