Answer: There are a number of perennials that will do well in. Most perennials also need a relatively large container compared to annuals because the perennials need to to gain some size and maturity before they will bloom with much of a show.
Having said that, here are a few perennials to try. For shade, hostas do beautifully in containers and are available in a wide range of foliage colors and textures. Astilbe will also thrive and bloom in a container of very rich, moist soil in partial shade. For a sunny location, you might try ornamental grasses, purple coneflowers, lance-leafed coreopsis, smaller daylilies and the taller sedums. Many of the herbs also do particularly well in containers.
There are zillions of annual plants and flowers that will thrive in containers - zinnias, cosmos, snapdragons, portulaca, geranium, hibiscus, etc.) and most vegetable crops. Container grown plants often need to be watered on a daily basis, so make it easy on yourself and use self-watering containers or hook up a simple drip irrigation system.
Let me recommend a great book on the subject: Container Gardening For Dummies by Bill Marken and the Editors of National Gardening Magazine (for more info, check out http://www.dummies.com). This takes you through the whole process of design, planting, care, etc.
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