The short answer is that prayer is deliberately, attentively, and openly turning your mind to God.
The longer answer is that there are in essence two kinds of prayers: meditative and intentional.
Meditative prayer is when one brings oneself to God with no explicit purpose beyond “simply” being with God–communing with God, listening to God. Meditative prayer is not about results; it’s purely about the experience of being with God.
Intentional prayer is the sort of interaction with God that most people mean when they use the word “prayer.” This is where one brings oneself before God with an end in mind; it’s what we do when we appeal to for God for help with a problem, or with a concern that we know we alone are incapable of satisfactorily resolving.
Generally, an intentional prayer will boil down to one of four core types: supplication (e.g., “Lord, I’m humbly asking you for this thing.”); contrition (e.g., “Lord, I feel so terribly bad about what I’ve done.”); intercession (e.g., “Lord, I’m asking for you to get directly involved, and turn this bad situation into something good.”); and finally pure, good old-fashioned gratitude (i.e., “Thanks, Lord, for looking out for me.”)
Need, remorse, helplessness, and gratitude. You can’t go wrong turning to God with either of those on your heart.
Books can be (and, goodness knows, have been) written about the reasons it’s spiritually, psychologically, and even materially beneficial to pray. (Prayer is beneficial materially because more prayer = less stress = clearer mind = more productive.) But the main thing to remember about prayer is that it’s an act that places you in your proper, natural relationship with God. In the best possible way, praying puts a person in their place—that is, it puts them in the best place anyone can be, which is before God with an attitude of humility, hopefulness, appreciation, and love.
In life, context is everything. You can’t know who you are, or what you’re doing, without understanding the context in which you’re living and acting. The great thing about praying is that it centers you at the great, humming, vibrant balance point between yourself, the created universe, and the infinite, infinitely compassionate power that created that universe.
Praying isn’t how you become someone better than you are. It’s how you remember who you are in the first place.