For gardeners who don't have the yard space (or don't have yards at all), growing plants in containers is the natural choice. Many garden plants can thrive just as well or better in pots and containers than they do an in-ground garden, and peppers are no exception.
Soil Needs
Annual peppers will do just fine in no less than three gallons of soil, but you will need more than that over time if you are growing perennial peppers, click here to find out more. Select a pot that has at least one drainage hole to release excess water, and fill it with potting soil. Most garden supply stores carry soil specifically designed for potted plants and raised beds. Either of the two will work fine, with potting soil being preferable over raised bed soil. Avoid using plain topsoil or dirt from the yard. I can say from personal experience that doing so will very likely stunt your plants and lower your chances of ever getting a harvest. Another option is to make your own potting soil (click here to find out how).
Watering
In a container or pot, your pepper plant will have different needs than it will growing in the ground. You will need to pay closer attention to watering, and fertilizer should be applied about once every two weeks. A general rule of thumb is that when the soil in your pepper plant's pot looks and feels completely dry on the surface, it's time to water. When watering, add water to the soil until it begins to drain out the bottom. Keep an especially close eye on potted pepper plants growing directly in the Summer sun. They can dry out and die in a matter of hours without watering.
Fertilizer
Choose a general purpose granular fertilizer or a granular fertilizer specifically made for tomatoes and/or other fruiting garden plants. Fertilize with about 1/4 cup of fertilizer about every three weeks, and very lightly mix or "scratch" it into the soil. Some other options for fertilizing include fertilizer spikes, which are highly convenient but expensive and sometimes difficult to find, and liquid fertilizer, which usually needs to be applied more often than the other choices.
Placement
Place your potted pepper plant somewhere where it will receive at least six hours of direct sunlight. Bear in mind that peppers enjoy the sunshine and heat, but somewhere like a concrete patio in the Summer afternoon sun may be a bit too much.
Pests and Diseases
In my experience, well cared for pepper plants very rarely suffer from pests or diseases. If a plant is well fed, well watered, and receiving the necessary amounts of light, pests and diseases are usually not a concern. Hot peppers in particular are very resilient towards insect pests. The main issue you will likely face is caterpillars. Peppers can be attacked by multiple species of caterpillar, but I have yet to see a plant overwhelmed by them. The easiest solution is to find, handpick, and kill or relocate the caterpillars. Wearing gloves or using tweezers is recommended, as some species of caterpillar can have venomous spines.