Nickel is a metallic element, number 28 in the periodic system. It is silvery in color and has a number of properties that make it attractive from a metallurgical point of view. It is quite corrosion-resistant and adheres very well to other metals, making it excellent for protective or decorative plating. It is also frequently used as an intermediate layer to improve adhesion between other metals, like when electroplating gold on silver, and as an alloying metal, like in many varieties of stainless steel and low-grade (less than 18 K) gold.
The problems when using nickel in jewellery stem from the one notable exception in the "quite corrosion-resistant" bit. It reacts very easily with a number of nitrogen compounds and unfortunately the amino-acids of our bodies are among them.
The term "nickel allergy" is, strictly speaking, a misnomer since the problem is more of a hypersensitivity. Your immune defense system is simply doing its job. It's just a bit "overzealous". Nevertheless, the problem is real and can become very acute. Once the "allergy" is triggered, the sufferer will react to much lower concentrations of nickel than before. Jewellery that was previously safe may become useless. Some will react to the nickel in coins, railings, cutlery and other household items as it gets dissolved by their perspiration and permeates into the skin. Not to mention the nickel that can dissolve from stainless steel sinks into the dishwater and further into the skin of any person sticking his/her hands into the water.
So what can you do? First and foremost you must avoid nickel in jewellery that's inserted in fresh piercings or in moist places. The issue is not so much if there IS nickel in a certain alloy. What matters is that it STAY there. Gold, for example, has a tendency to "bind" nickel so that down to 18 K a small amount of nickel is usually "safe", except for "white" gold which often contains (and releases) too much nickel. When you get down to 14 K, the nickel released is often getting dangerously high, at least to those who are already sensitized. Stainless steels frequently contain nickel. The "hypoallergenic" varieties are frequently simply those that "bind" their nickel so thoroughly that little or none of it is released.
In extreme cases the sufferer can find even the most pure metals impossible to wear. Even 24 K gold (nominally 100%) can contain traces of nickel or other contaminations, but this is extremely rare. Most find a marked improvement in simply going from 14 K to 18 K. Other metals are now finding their way into the jewellery business, titanium and niobium rapidly gaining a well-deserved reputation for safety. The oxide layers on their surfaces are sufficient barriers against corrosion and wear and they also do not have nickel's tendency to bind into the cells of the body and trigger the immune defense system. Otherwise, inert plastics (Nylon, Teflon) work well. Some have reported successful experiments with "alien" organic objects like wood, ivory or porcupine quills which don't contain anything that can dissolve into the body, but care should be taken with these since they can trigger your immune defense system in their own right, being (formerly) "living" materials themselves.
Other metals, like chromium, trigger similar effects like nickel, but less strongly so, in most cases. However, many nickel-free alloys contain very high amounts of chromium instead and hence they, too, may release enough chromium to trigger "allergic" reactions.