The best flowering plants for your patio depend on how much light the patio gets. There is a difference between partial shade and no sun at all. Also, consider the size of your containers. You don't want plants that may have too large of a root system for the pots.
Containers look nice with a variety of plants. Consider adding something tall for vertical interest as well as a trailing plant that will hang over the lip of the pot. A tall container might not look right without a tall plant for visual harmony. Make sure that the plants you include in one container have similar watering needs. For instance, you do not want to mix succulents with water-loving plants.
Hydrangeas have become popular container plants if the plants are small enough for the pot. Make sure they have room to grow. In fall, you can take them out of the pot and plant them in the ground, so they live through the winter. Hydrangeas blooms last a long time so will look nice for a long time.
Impatiens, begonias, lobelia, some salvia, and fushsia all do well in partial shade. It would be best for them if they get their sun in the morning as afternoon sun can be so harsh. Many types of begonias produce flowers and many that offer interesting foliage. Consider foliage plants with colorful leaves like coleus and caladium.
When you visit your local plant store, look at the tags to check sun needs of each plant. Remember that potted plants need to be watered more often than plants in the ground because they dry out faster. Also, heavy watering can leach nutrients out of the soil so you will want to feed them more often than you do plants in the ground. Mulch the container to help retain moisture.