The speaker
The speaker is reflecting upon the value of making the most of one's day and taking the world by the reins rather than letting the "rain[y]" days in life discourage him. The speaker suggests that having an optimistic view of life is what ensures a "happy" life. Those that are "secure" in their hope and happiness will experience the true "joy" of life.
The vicissitudes of life
The poem positions life, especially its caprices, against the everyman, describing the ups and downs of life (and often personifying them) as forces that may influence one's sense of fulfillment and happiness. The speaker's argument against his opponent is that one who lives in the now, accepts one's past, and does not fear one's future is capable of triumphing over life's uncertainties.
The "happy" man
This unnamed hypothetical character is a reference to the average person. Dryden argues that happiness is attained by one who can appreciate the now and have a consistently positive and grateful attitude towards life, regardless of the situation.