Blossom End Rot
Despite the name, this plant problem doesn't really result in plants or fruit rotting. It's also not caused by a virus, bacterium or fungi as is often suggested by gardeners who haven't encountered it before. Blossom end rot is in fact caused by a nutrient deficiency - calcium deficiency to be precise.
Plant calcium deficiencies may be the result of a lack of calcium in your soil but this is actually relative uncommon. The most common causes are inconsistent watering, a high concentration of salts in the soil, planting plants in cold soil and/or heavy clay soil (because in such conditions plants don't produce a robust root system that can seek out water and nutrients during periods of stress and lack of water) and damaging plant roots by digging too close the plant for instance. Each if these things can prevent plants from absorbing enough calcium even when sufficient calcium is available in the soil.
How It Spreads
Because blossom end rot is not caused by a pathogen, it does not spread from plant to plant.
Tomatoes, capsicums and other members of the nightshade family are most commonly affected by blossom end rot though other fruiting plants, such as squashes, may produce similar symptoms. Many other plants can suffer from calcium deficiency but the resulting disease often goes by a different name. For instance, bitter pit in apples is the result of calcium deficiency as are tip burn in brassicas and lettuce and blackheart in celery.
Blossom end rot results in black spots at the base of tomatoes - that is, the end where the flower was. In mild cases tomatoes may just have one or more small spots but in severe cases, up to half of a fruit's skin may turn black and harden. The fruit is still edible however.
Once symptoms appear, the damage cannot be reversed though remedying the cause of the calcium deficiency can prevent damage in later fruit.
In the unlikely event that soil is deficient in calcium, a calcium preparation can be applied to the soil near the plant roots. A calcium chloride spray can also be applied to plant leaves to quickly provide calcium until the plant roots have a chance to take up the calcium added to the soil. Don't do this often though because too much calcium chloride can be toxic.
If a period of drought has caused the problem, water more frequently, mulch well and incorporate some organic matter into the soil. If you have been applying lots of high nitrogen fertiliser, stop. This can not only prevent calcium uptake but it will also reduce flowering and therefore the amount of fruit that you will get.
Checking the pH of your soil is also a good idea. It should be roughly 6.5 for tomatoes and other nightshade vegetables. If it's too acid or alkaline, the calcium available in the soil will be harder for plants to absorb so add lime (for acidic soil) or sulfur (for alkaline soil) according to the instructions on the packet.
Commercial tomato growers sometimes spray foliage with a calcium chloride spray to prevent blossom end rot but the best prevention is to ensure constant watering of plants and prevent salt build up that may inhibit calcium uptake. In areas of high rainfall, planting tomatoes in raised beds can help prevent blossom end rot due to overwatering. Transplanting into warm soil can also help prevent the disease. Incorporating organic matter into the soil prior to planting will help the soil hold moisture in dryer climates.