How many Hindu gods and goddesses are there?

Hinduism is a religion, which enables us to lead a original human life in spite of so many beliefs. This is possible because Hinduism associates each of the basic forces with a god.

The Basic gods:

The basic forces and the associated Hindu gods are,
1. Dreaming force - Indra
2. Illusory force - Shakti
3. Voluntary force - Brahma
4. Semi voluntary force - Vishnu
5. Involuntary force - Shiva
6. Status altering force - Shani
7. Compulsive force - Yama

Brahman - The supreme god

Above these seven forces is the basic force selector called Brahman, whom we call the supreme god. Thus, use of Brahman gives us a feeling that there is just one god.

The principle of Hindu gods:

The basic forces are associated with the opposite element in nature. Thus, Compulsive or Yama force is associated with death. Brahma, the transient for is associated with creation of eternal universe.

The Incarnations of Vishnu:

Vishnu force is our default force and thus our first choice force. This force generates a sinusoidal or wave like graph. Each part of the wave generated by this force is associated with an Incarnation of Vishnu. Thus, Vishnu has ten popular Incarnations and many other not so famous incarnations. In each incarnation Parvati too incarnates thus increasing the number of gods. In Ramavatara, there is the popular god Hanuman who is a Shakti god.

The central three basic forces viz. Voluntary, Semi voluntary and Involuntary forces are very difficult to use in pure form. Therefore, we have an intermediate stage called consort. Saraswati, Lakshmi and Parvati are consorts of Brahma, Vishnu and Shiva respectively.

Using the Involuntary Shiva force is difficult in spite of having an intermediary state or consort, Parvati. Thus, we have two two children Ganesha and Subramani.

The Illusory or Shakti force gives an illusion of strength. Thus, gods based on this force are common in villages all over India. This greatly adds to the number of Hindu gods.

Thus, Hinduism needs many gods to enable a Hindu to lead a original human life in spite of so many beliefs.

Another View:

Actually many people in ignorance of Vedas say that Hinduism has many gods while Hinduism is a monotheist religion.

Vedas says:

Na dvitityo Na triyaschthurtho naapyuchyate|
N a panchamo Na shshtah sapthmo naapyuchyate|
Nashtamo Na navamo dashamo naapyuchyate|
Yagna yetham devamekavritham veda
Sa sarvassai vi pashyathi yachha praanathi yachhana|
Tamidam nigatam sah sa yesha yeka yekavrideka yeva|
Ya yetham devamekavritham veda
Atharva 13.4[2]19-20

There is no second God, nor a third, nor is even a fourth spoken of

There is no fifth God or a sixth nor is even a seventh mentioned.

There is no eighth God, nor a ninth. Nothing is spoken about a tenth even.

This unique power is in itself. That Lord is only one, the only omnipresent. It is one and the only one.