Preservatives must exist in skincare (and other products) as they kill microorganisms and water borne bacteria as well as preventing the growth of bacteria, yeast and mold. Some preservatives, however, are very harsh in how they preserve the shelf life of a product and yet there are others, which can do the job without seriously affecting your skin (or, for that matter, the environment). Many of the top-class skin care products do contain reactive ingredients, which, without proper care, would deteriorate and so make the product basically, useless.
A very effective preservative are the molecules found in nature, in plants such as pine resin, rowan berries and willow bark. These are non-paraben, non-formaldehyde (a really nasty product) and non-isothiazolone based preservative system. Such include Benzyl Alcohol, Salicylic Acid, Glycerin and Sorbic Acid. Note that Salicylic Acid is also a very effective cleanser for acne-prone skin, though when used as a preservative it is not usually in a high enough concentration. There is also a complication in that in most personal care products, you're not always going to find the natural version of Benzyl Alcohol. There is a synthetic version, made in the laboratory by mixing benzyl chloride with sodium hydroxide. This is not as "natural" and benzyl chloride can be irritating to the skin.
Dehydroacetic acid (DHS) is also used as stabiliser for cosmetic and pharmaceutical products due to its fungicide and bactericide activity. This is also usually combined with Benzyl Alcohol.
Commonly quoted as preservatives (though they are not!) are Vitamin E, rosemary extract and grapefruit seed extract. If a product contains water (and this includes hydrosols, floral water, aloe vera, goat’s milk - all of which contain water) or will come into contact with water (for example, a scrub used with wet fingers) a preservative is essential to help prevent microbes growing. Preservatives stop growth by acting on spores when they germinate and killing cells (usually by disrupting cell membranes) or by making the product hostile to growth.
Many products contain a range of preservatives and there is a reason for this, which is simply to make sure that as many of the bases as possible are covered. Few, if any preservatives are multi-functional (i.e. there is not one preservative that will do the whole "preserving" job on its own) so you will often find a range of low-concentration preservatives in many products.
For further research and reading on this subject, I suggest that you visit makingskincare.com which has an excellent guide to preservatives in general and dispels many of the myths surrounding their use.