From a biochemical standpoint, vitamins are small organic molecules that act as cofactors whose presence is required for certain proteins and enzymes to function. To put that into more common terms, vitamins can be compared to common work tools that humans use. For example, although you may have the strength to turn a screw, you are unable to grip, turn and sink a screw without a screwdriver. Hence, proteins in your body, which are responsible for the upkeep and health of your body, may possess the 'strength' to perform a much needed function like extract energy from food, however, it cannot without the use of a vitamin which helps the protein 'grip' its substrates. Our body is incapable of creating new vitamins, thus the only way to replenish lost or broken vitamins is through our diet. Our body's ability to grow and continue functioning properly is directly limited by the intake and presence of vitamins.
There are 13 vitamins each identifiable with a letter and possibly a number. The names, functions, sources, recommended daily allowances and resulting diseases caused by deficiency are all mostly accurately summarized in a table which can be found on wikipedia:
Instead of focusing on the nitty-gritty details about the biochemical functions of each vitamin here, I encourage you to read all about it on wikipedia. Something I do wish to dwell on is the correlation between diseases and vitamin deficiency. There are several very well documented diseases such as scurvy (gyyarr) which arise as a result of short-term vitamin deprivation (ie the amount of time it takes to sail across the ocean). Diseases such as these are becoming more of a rarity in today's society since most people usually take in enough vitamins to keep such disease at bay. What some have wondered, however, is if prolonged consumption of a 'western' diet which is not as rich in vitamins as fruits and vegetables are could be responsible for a host of other more common diseases that many in our society are encountering today (ie cancers, autoimmune diseases....). While scientific studies have yet to definitively establish a correlation between something as broad as a lifetime of slight undernourishment and cancer for example, the internet is rife with anecdotal evidence supporting this conclusion. A prominent and entertaining example can be found in the recent movie "Fat Sick and Nearly Dead" (http://www.fatsickandnearlydead.com). Personally, I think that loading your diet with more fruits and veggies couldn't do much harm, however, I don't believe that doing so is going to be the end-all of diseases as many would like to believe.