Prayer is usually sorted by what it does, petition, praise, thanksgiving, confession, intercession, but the more revealing division is by form, because form is what a tradition actually teaches and enforces.
Fixed prayer is set text at set times. Muslims pray five times daily at hours determined by the sun, facing Mecca, in a sequence of standing, bowing, prostrating and sitting that is identical in Jakarta and in Lagos. Observant Jews pray three times a day from a fixed liturgy, and the obligation attaches to the hour rather than to the mood. Christian monastic communities divide the day into offices and rise in the night for one of them. The point of fixity is precisely that it does not depend on wanting to.
Free prayer is composed while it is spoken. It dominates Protestant and especially evangelical practice, where praying aloud extempore in a group is a normal acquired skill and reading a written prayer can be heard as insincere. The two forms have been in real tension rather than polite coexistence: the English quarrels over set liturgy in the sixteenth and seventeenth centuries were about exactly this, and people were imprisoned over it.
Silent prayer dispenses with words. Quaker meeting waits without any programme at all. Orthodox hesychasm repeats one sentence until it sinks below deliberate thought. Catholic contemplative prayer aims at wordless attention and has a technical literature on how to tell that from mere absence of mind.
The body is not incidental to any of it. Muslims prostrate; Orthodox Christians make full prostrations in Lent; Jews bend at named points of the standing prayer and rise on the toes at others; Hindus touch the feet of an image or a teacher; Tibetan pilgrims measure a route in body-lengths. Where a tradition specifies what to do with your knees, it is making a claim about what prayer is.