Pepper plants, like their relatives (tomatoes, eggplants, etc) do well in summer heat, provided it is not too intense. The plants should be set out (after a generous hardening-off period, of course) after frost is ended for the season and the soil has had a chance to warm--two to three weeks after the nominal last-frost date is about right. Hot types are generally a bit more toleant of cool conditions than the sweet types, and could be set out a bit earlier.
Pepper plants respond to a fairly high level of soil fertility, though if the ground is too rich, theyt may grow lavishly and take their time flowering. The ground should be well-drained yet moisture retentive, which may sound paradoxical but isn't if lots of compost or well-aged manure have been added to the soil. A mulch helps stabilize soil temperature and conserves moisture. Plants should be kept well-watered, but not over-watered. Occasional light wilting in the afternoon isn't harmful but is the signal that it is time to water.
When the plant has reached sufficient size, it will start to throw chaste, waxy blooms that are modest, yet attractive; usually in white, but on some varieties a medium lavender and often in considerable numbers. When the blossoms drop, the incipient fruit remains. This enlarges in size, eventually growing to become the sweet or hot pepper we've chosen.
Pepper plants are subject to many of the same pests and diseases as tomatoes and eggplants, but in our experience do not succumb as easily as either. Flea beetles haven't bothered them in our garden (perhaps because they were too busy devouring our unprotected eggplants!), but occasionally the tomato hornworm (larval stage of the sphinx moth) does some damage. These are easy enough to pick off; wear gloves if you're squamish.
The peppers are ready to harvest when they've reached the stage the gardener is seeking--sweets or hots may be harvested green or ripe. By all means wear gloves when harvesting the hot ones, and be sure to wash before touching your eyes.
Intense summer heat (consistent temperatures above about 95 degrees Farenheit) will cause the blossoms to drop without fruit-set. The plants endure, however, if kept well-watered, and begin bearing again when the heat abates. In our gardens, peppers are typically the last member of the family to yield in the fall garden, bravely carrying on after the tomatoes have surrendered to the cold of autumn. Many can survive a (very light) frost.