Modern Hinduism

Hinduism is difficult to pin down as a religion; with about 330 million gods, trying to classify its practitioners gets out of hand quickly. Nevertheless, there are some beliefs shared by most Hindus, and such beliefs provide a good introduction to understanding the faith. (This list is adapted from http://www.himalayanacademy.com/basics/nineb/.)

Hindus believe in a one, all-pervasive Supreme Being, Brahman, who is both immanent and transcendent, both Creator and Unmanifest Reality.
Hindus believe in the divinity of the four Vedas, the world's most ancient scripture. These primordial hymns are God's word and the bedrock of Sanatana Dharma, the eternal religion.
Hindus believe that the universe undergoes endless cycles of creation, preservation, and dissolution.
Hindus believe in karma, the law of cause and effect by which each individual creates his own destiny by his thoughts, words and deeds.
Hindus believe that the soul, or atman, reincarnates, evolving through many births until all karmas have been resolved, and moksha, liberation from the cycle of rebirth, is attained. Not a single soul will be deprived of this destiny. Unlike in the West, salvation is not the attainment of eternal life; rather, it is a complete and simultaneous rejection of the self and union with the entirety of the divine.
Hindus believe that divine beings exist in unseen worlds and that temple worship, rituals, sacraments and personal devotionals create a communion with these Gods.
Hindus believe that an enlightened master, or satguru, is essential to know the Transcendent Absolute, as are personal discipline, good conduct, purification, pilgrimage, self-inquiry, meditation and surrender in God.
Hindus believe that all life is sacred, to be loved and revered, and therefore practice ahimsa, non-injury, in thought, word and deed.
Hindus believe that no religion teaches the only way to salvation above all others, but that all genuine paths are facets of God's Light, deserving tolerance and understanding.

To reiterate, not all Hindus believe all of these. For a start, Hinduism itself doesn’t much care about classifying its followers—to do so is a very "western" attitude. And beyond this, Hinduism also admits paradoxes and contradictions, which to a westerner steeped in rationalism, seems to weaken the faith. Hinduism does not view paradoxes as problematic: one can be an atheist, and still be a Hindu. While it is impossible to explain this in the language of logic westerners so love, intuitively understanding the nature of Hinduism's tolerance for contradiction is one of the first steps to a true conception of the faith.

Three (or Four) Ways to Salvation through Yoga

In Hinduism, three major routes to salvation exist. In contrast to western belief, in which all followers of a god must obey the same rules to achieve salvation, Hinduism permits many paths to moksha.

Karma Yoga: This is the yoga of duty; it is also called Dharma yoga. Practitioners of Karma yoga follow and fulfill their roles in life to the best of their abilities. One of the key points of this yoga is that a person must accomplish his or her assigned tasks without focusing on the outcome; to do so is to be vain. By leading a life compliant with dispassionate duty, a follower is able to achieve good karma, which leads to re-birth in a higher caste. For more information on the caste system, which has much to do with Karma yoga, click here.
Jnana Yoga: Also called the yoga of knowledge, practitioners of this yoga focus less on worship and more on philosophical study. The ultimate goal of this yoga is to achieve an understanding of the connectedness of one’s soul to Brahman. Once this fact is known, followers can achieve moksha, because they realize the transience of the physical, and are able to unify with the eternal divine.
Bhakti Yoga: This is the yoga of devotion, and it is the most commonly practiced yoga. Followers of Bhakti yoga achieve salvation through dedicated worship of a god or gods, offering sacrifices, singing hymns, dancing, and meditating on a god(s) works. This appeals to many Hindus, because it offers a way to achieve moksha without necessitating numerous rebirths—a form of “instant” salvation.
Raja Yoga: Although similar to Jnana practices, some consider this a separate yoga. Raja yoga is similar to Buddhism. It emphasizes meditation and detachment from the physical, in order to achieve enlightenment. The difference between Jnana and Raja yoga, for those who see them as different, is that Jnana yoga focuses on study and thought, whereas Raja yoga focuses on achieving a sudden, unprompted realization of the impermanence of the physical through meditation. They both have the same goal, but try to achieve it through different means.

Sources:
http://www.himalayanacademy.com/basics/nineb/
http://www.swamij.com/four-paths-of-yoga.htm