In the broad scheme of trinitarian doctrine, there are only three overarching positions to choose from, each of those three being able to be further divided into different variations. These three options are modalism, tritheism, and subordinationism; there are no other alternatives, and every view on the Trinity fits somewhere within these categories.
All three systems broadly agree on the three basic facts that there is one God, and three divine persons of the Father, the Son, and the Holy Spirit. But these facts alone, stated this way, are too vague; and the way each system explains how these facts fit together is different. They do not agree on what it means that there is one God, or what it means that there are three divine persons.
Modalism explains monotheism by arguing that there is only one divine person, and thus only one God. It either makes the three persons out to be one person, or else denies either the divinity or the distinct existence of two persons. Sometimes this is done by denying distinct existence of the Son and Holy Spirit, other times by saying that “Father”, “Son”, and “Spirit” are just three different names, or three different modes of manifestation, of one person, other times by declaring that the three persons are ultimately a single person at the deepest level, although on the surface and in a relative way relate to each other as though three persons. Thus by defining the oneness of God as there being only a single divine person, they ultimately deny that there are three divine persons in anything but name only.
Tritheism goes to the opposite extreme by denying that there is truly one God by making the three persons not only really distinct, but also separate, and entirely equal. By proclaiming three independent identical divine persons, they make there out to be three gods. A weak attempt to say otherwise often comes in the form of arguing that there being one God simply means that there is only one divine nature of Godhood, which is shared by the three identical persons. But this falls apart easily, for just as three human persons with one common human nature are three men, so the tritheistic reckoning of three divine persons with one common divine nature makes there out to be three gods.
Subordinationism avoids the pitfalls of modalism and tritheism. There is not one God because there is only one divine person, as there are three divine persons, truly distinct from each other. It likewise avoids the pitfall of tritheism by not making the Son and Spirit identical and equal to the Father, but rather regards them as subordinate. There are various forms of subordinationism, but in the most rational form of it, and that taught by scripture, the Son and Spirit have the same divine nature as the Father, and are thus equal to Him in respect to Godhood; but They are subordinated to Him as Their Cause and Authoritative Head. Thus, in this classical trinitarianism, there is one God because there is only one Supreme uncaused Cause of all, and one Supreme Authority over all, the person of the Father. Not only is all creation caused by the Father through His Son and Spirit, but His Son was atemporally begotten of Him before the ages, and His Spirit eternally proceeds from Him; thus all things run up into one supreme cause, the Father, Who alone simply is what and who He is without cause, source, or origin. Likewise although the Son has been given all authority in heaven and earth, even He Himself is subject to the Authority of the one Who subjected all things to Him, His God and Father. Thus all authority runs up into one Supreme Authority over all Who has no higher authority above Him. Thus there is one God, the Father, and yet there are three truly distinct divine persons.