The wasp is a member of the Hymenoptera order (which also includes bees and ants). Species belonging to this order generally employ an ovipositor to lay eggs (see the movies Aliens for an enlarged version). In wasps, the ovipositor is no longer used for this function, instead it is used to sting.
The ovipositor is a sheathed tube which normally rests in the abdomen of the wasp. It consists of two, what look rather like, barbed swords called the lancet and stylus; a poison sac and the muscles to move the whole apparatus. The lancet and stylus do not thrust into the victim, rather they move in a circular motion like the blades in a Swiss Army Knife.
One of these cutting tools, say the lancet, will anchor itself in the flesh of the victim (using its barbs) while the stylus digs in deeper. Then the stylus will anchor while the lancet digs. This continues until a large enough wound is created into which the venom is pumped. The whole process takes less than a second.
The barbs are small to enable the wasp to disengage itself and fly off unharmed. A bee does not have this luxury. It has larger barbs which cannot be dislodged by the bee, who subsequently dies.
The wasp venom consists of various proteins and enzymes. It also contains an acetylcholine-like substance, histamine, serotonin and a kinin (peptides which cause slow muscle contractions and lower the arterial blood pressure).
In most people the wasp sting causes an initial sharp localised pain followed by some swelling and itching. The itching may be alleviated by applying antihistamine cream. Alternatively, vinegar can be used. To reduce the swelling apply a cold compress.
However, a small percentage of the population are at risk of suffering an allergic reaction to the venom (known as a systemic hypersensitivity reaction). This reaction ranges from large swelling to sudden death by anaphylaxis (similar to a heart attack).
The wasp sting first aid kit for people who are allergic to wasp stings would include adrenaline (with syringes), antihistamine and a ventolin inhaler.
Wasps generally will not sting a human unless they feel threatened or if someone wanders too close to their nest. If they land on a human, they will usually fly off again if unprovoked.
Disclaimer: Please don't use this w/u for medical advice, it is derived from the sources quoted below