To allow the plants to grow best, space them so that they won't compete with each other for water and soil nutrients. This will also allow air to circulate between the plants better, which is important in preventing diseases. Cosmos, zinnias and tuberous begonias are particularly susceptible to mildew and require proper spacing.
Space your annuals according to the specifications on the plant tag or seed packet. They will grow to fill in the empty space. Often the tag gives a range for spacing, such as "plant 8 to 12 inches apart." Northern gardeners should stay to the lower end of the range or even slightly below it in order to achieve a lush-looking display by midsummer, before the growing season starts to wind down again.
You can wedge annuals more closely in containers—divide the recommended spacing in half.
When you plant your annuals, loosen the roots. If the plants are very root bound, you may want to slice into the root ball with a knife. Remove some stems as well to compensate—you want to maintain the root to plant ratio to avoid stressing the plant. You can also remove some flowers and buds when you plant them, to encourage the plant to use its energy to send out roots.