What are the three forms of brahman?

In the Bhagavad Gita, Krishna says: "Whenever a devotee wishes, with unwavering faith, to worship me in a particular form, I take that form."

There is no limit to the forms that Brahman can take.

If you are, however, referring to the Hindu Trinity, or Trimurti, the forms you are probably thinking of are Brahma, Vishnu and Shiva. Formerly when Hinduism was more Pantheistic, the Trinity consisted of Agni, Vayu and Surya. This is sometimes called the Cosmic Trinity, I believe, representing fire, wind, and light.

Actually, the personified manifestation of Brahman is Lord Vishnu. It is believed that when the last Manavantara ended, the whole universe was destroyed and the only entity that survived was Lord Vishnu (personification of Brahman/personification of the syllable OM or AUM), who then created Lord Brahma for creation and Lord Shiva for destruction and preservation. While he (Lord Vishnu) took the work of looking after the universe i.e. sustaining. Lord Vishnu has the most power among the 3 forms, as he is Brahman himself. 'Vishnu' means the all pervading one, the one who possesses everything in this universe. While Lord Shiva has been given most of the work, Lord Vishnu takes birth on earth (bhur-loka) in times of utmost crisis.

In this present age of Kali-Yuga, Kali (devil in Hinduism, spelt as kuh-lee and not kaa-lee referring to the Goddess Kaali) is the son of a Brahmin. He is the negative manifestation of Vishnu. Keep in mind that Vishnu is Brahman, devoid of any Guna (neither saguna nor durguna i.e. nirguna). So, in order to clean the dirt, he will take birth in the form of Lord Kalki to annihilate him.

Lord Vishnu has always sent his incarnations in the form of ansh-avatars in mortal bodies. Note this that he hasn't come on earth himself (Mohini avatar was only an illusion, as she appeared twice but did not take birth in a mortal body).

We, as humans, are just the media. We think that God has given us free will, but the truth is that "we do what we are born to do...and not what we want ourselves to do". This concept has a deep meaning and many people would find this complicated to understand.

Why did Lord Raam take birth and went through time consuming processes like birth, childhood, youth, marriage, fatherhood and later death? If Lord Vishnu took birth just to kill Ravana, why did he risk his wife Sita's life? Why didn't Lord Hanuman just trample on Ravana and finish him off?

The concept of Karma (deeds) comes here. Everyone is born to do a task i.e. a doctor has to practice only medicine, but not a mason, even if he knows the composition of concrete and cement.

It was pre-prophecized that Ravana had to die at the hands of Lord Raam and no other.

In Kali Yuga, Lord Vishnu took birth in form of Gautama Buddha. Now, why did Buddha teach against Vedas? Because of the necessity at the time being. The same entity who took birth in the form of Matsya-avatar to preserve the Vedas, taught against them in Kali-Yuga. This was because the increasing influence of Kali/devil, the teachings of Vedas have very less attention and hence, humanity should find inner peace via meditation and some rules.

I mentioned before that everyone has a destined task: be it a doctor, or a terrorist. Sufferings are a part of every being's life.