The main philosophy linked to Hinduism is Vedanta, the philosophy of monism. Monism, not to be confused with monotheism, is the belief that there is only one thing in existence, that all that is is part of one whole, which is the entire contents of reality. This doesn't mean that Hindus can't distinguish a table from a telephone, it means that below the surface, on the quantum level of reality, they believe that all is connected, all is really one. Modern science doesn't exactly disagree, either.
All that is is One. That fundamental Whole, the entirety of reality is what we commonly hear reffered to as God. Even in Christianity, where people tend to personify the concept of God and "trap" God in human form, there is the basic concept that God is 1. Omniscient and 2. Omnipresent. That means that whomever we refer to as God certainly isn't limited to a single human form. God must be everywhere always. This is why Muslims consider it a huge sin to draw or create representations of God, because it betrays what God really is- everything, everywhere, always. Although Hinduism is the only explicitly monist of these three, at their core they agree. Hinduism is monotheistic and monoist. Islam and Christianity are monotheistic, and hint at the concept of monism. Islam integrates that concept fluidly in its Sufi branch, which is ecstatically monist. It is amazing how diverse the cultural wrappings are, and how one and unified the essence is!
And it is to this cultural wrapping we will speak next. Although monotheistic and monist at heart, undoubtedly there are a number of Gods and Goddesses in Hinduism. How can this be? Well, how can all of us exist in a world that is just one Whole? These Gods and Goddesses each embody a different aspect of the fundamental Reality, a quality of God. Each one is a piece of God. Each of these representations has a long set of allegorical stories, mantras, songs, images. Each God has a Shakta, a Goddess. They come in pairs, like Yin and Yang. Hindu families will worship one particular aspect of God or pair over all others, that is their Ishta-devata. The purpose of this whole confusing concept is to allow the worshiper to 1. Learn values and spiritual qualities from the stories, 2. Form a personal relationship with their form of God, to be able to Love God.
It is hard for some to feel devotion to an indescribable reality, while loving and honoring a representation of that Reality in the form of human incarnation, through teaching-stories, songs, mantras, and festivals is easier for most people to live.
There is a time in our spiritual evolution to go beyond the forms, the representations, the limited image of the Indescribable. However, the Ishta-devata concept serves many well, for many lifetimes before they are ready to "go beyond". In fact, in Christiany, Jesus Christ is an Ishta-devata, considered an incarnation of the Divine. Through loving Jesus, by hearing his teachings and story of his life and sacrifice, through singing songs, Alleluiah and Amen, people form a personal, devoted relationship to the Divine. Islam, most recent in the trilogy of the Abrahamic religions, does away with this concept- Jesus is a revered prophet, not embodiment of God, and this is the difference and root of rivalry between the two.
Interestingly, Buddhism teaches the same concept we have defined as God- the existence one unified Ultimate Reality manifested in the multitude forms, but uses no term God. The Buddha taught that to try to describe the indescribable and to label what is far beyond labelling was simply not to be done. One can only experience Ultimate Reality through very deep meditation, but words were hollow in trying to convey it. So experience, not labelling or describing, is the Buddhist path. Many think that belief in God is not possible in Buddhism, but it is there, pointed to and not trapped in forms or labels. This really agrees with the Islamic value of not limiting God by representation in any image or form.
Ishta-devata is very powerful and useful to so many people, helping them to strengthen their faith in Ultimate Reality and to practice their values dilligently, to be the very best and highest they can. It is the heart of Bhakti- devotional practice of chanting, loving and trying to see the Beloved everywhere in everyone and everything. In Bhakti, ones personal relationship with God becomes ones personal relationship with life itself, loving until all distinction between lover, beloved and loving merges into wholeness, Yoga (union/wholeness).