This topic is huge, of course, I encourage you to review any number of textbooks and other introductions in print or on the web, looking especially for the terms "literary theory," "critical theory," and "critical approaches to literature." I have included a couple of internet links below.
Most of the approaches are from the twentieth-century, so I'll concentrate on a few of these modern approaches:
New Criticism (c. 1920s+) seeks to understand how the text is structured as a self-standing artifact and how it develops its own meaning (through symbols, themes, etc.). This approach emerged as a challenge to biographical approaches to literature, in which the life and statements of an author were used to interpret the literary text.
Psychoanalytic criticism (c. 1900+) seeks to understand how the text can represent the individual's mind (illustrating concepts such as repression, sublimation, etc.). Archetypal criticism, which is often viewed as an offshoot of psychoanalytic criticism, seeks to uncover supposedly universal symbols of the collective mind (or "collective unconscious") of humanity.
Structuralism (c. 1960s+, but prefigured in the Russian Formalism of earlier decades) seeks to uncover the large patterns in a text and the connections between that text and some larger system. Deconstruction is a reaction to structuralism, pursuing the goal of uncovering large patterns and connections only to undermine the very concept of stable patterns and connections.
Feminism (c. 1960s+, but prefigured as well in earlier decades, especially from the later 1800s onward) seeks to uncover the ways in which literature represents (explicitly and implicitly) what it means to be a woman. Feminism is really a grouping of multiple approaches and philosophies, including radical feminism, social constructivist feminism, third-world feminism, eco-feminism, and others. Lesban and gay approaches (as well as the more recent developments in queer theory) are in many ways an offshoot of this branch of critical approaches to literature.
In this post, of course, I've hardly provided an extensive listing of even the recent developments in critical approaches to literature. I can say with assurance, though, that no one approach is "the most successful" of them all. Each approach has its particular strengths and limitations. Pychoanalytic approaches tend to work very well with text from the era of Romanticism, for example, but have more limited application to many more recent literary works.
The best practice in literary theory, I believe, involves understanding the strengths and limitations of the various approaches, selecting approaches that work well with the particular works that you are reading, and even combining approaches in new and meaningful ways.