Psychodynamic Theories are descendants of the original psychoanalytic approach developed by Sigmund Freud in the late 1800s. Dr. Freud was one of the first "psychotherapists" (professionals who treat mental problems with a talk therapy) and was nothing if not influential. Freud introduced the idea that the mind is divided into multiple parts, including the irrational and impulsive Id (a representation of primal animal desires), the judgmental super-ego (a representation of society inside the mind), and the rational ego which attempts to bridge the divide between the other two parts. He popularized the idea that the mind has conscious and unconscious parts which can conflict with one another, producing a phenomena called repression (a state where you are unaware of certain troubling motives or wishes or desires). His basic therapeutic idea was that mental illness was caused by mental tensions created by repression, and that mental health could be restored by making repressed knowledge conscious. As it turns out, reality is more complicated than this. Talking about your problems and coming to understand them doesn't necessarily make them go away, but it can be very helpful nevertheless. Many ideas from psychoanalysis turn out to be important, including the idea of repression (and the related idea of dissociation) which has developed into the study of coping strategies and defense mechanisms (ways that people attempt to manage or ward off knowing about stressful information).
One branch of modern psychodynamic theory (sometimes called "object relations" theory) is much less concerned about struggles between parts of the mind, and much more concerned about how people understand and represent their relationships with other people. The "objects" in object relations theory are representations of people (how others are experienced, represented and remembered by the person doing the objectification). Object relational therapists note that people's early relationships often set the tone that later relationships will take. This occurs in part because of a phenomena called transference, and also because what you experience early in life seems "normal" to you and you become in some ways drawn to new relationships that help you replicate that original "normal" feeling. This tendency works out well when early relationships are healthy, but very poorly when they are disturbed. People whose early relationships involve abuse or neglect often end up not feeling quite comfortable in later relationships unless those relationships recreate in some fashion those early abusive or neglectful dynamics.
Transference occurs when people use representations of older relationships as a means of jump-starting their understandings of new relationships. When an older relationship has been "transferred" onto a newer one, the older relationship will be the point of comparison against which the newer one is judged. The person doing the transferring may read characteristics or tendencies into the newer relationship that aren't there, simply because they were there in the older original relationship. For example, a young man who has had a difficult and distant relationship with his father, might generally react angrily towards other adult males, but not really have insight as to why he does this. He might end up sabotaging career prospects by alienating potential employers if he can't get a grip. Object relational therapists might help this man by making him aware of his prejudicial pattern, helping him to process his anger feelings in the moment (should he attack the therapist), and by offering a new model of what a relationship can be like (e.g., trusting, trustworthy, non-abusive) which the young man can then transfer to other relationships.
The key insights to take home from psychodynamic theory are: 1) that the mind is not so straightforward as it might seem, but instead may play tricks on you (e.g., you may end up repressing knowledge because it is disturbing to you), and 2) that your early relationships set the tone for your adult ones. Psychodynamic techniques are designed to help you become more conscious of unconscious habits, patterns and preferences laid down early in your life that may not serve you as an adult. As you become more conscious, you may gain the motivation to change those no-longer-working habits and tendencies Psychodynamic approaches help create insight. Insight is a motivator; it is not a cure. If knowing "why" you have a particular problem is important to you; if it will help you become more free to challenge what you are presently doing that doesn't work for you anymore, then learning more about psychodynamic techniques is a good idea for you.