In appreciating [full understanding being impossible due to non-match between our finite nature & finite experence on the one hand, and between the infinite nature & infinite embraciveness of God and the Trinity on the other] the concept of one God, but three Persons, it is essential to realize:

1.  Difference between Nature & Personality.
     God, angel, human, etc. each has a Nature special to itself and to others sharing that Nature;
     but each individual within & sharing that Nature has a special distinction termed Personhood/Personality.

2.  Person = someone having a situational framework whereby the unique    sum-of-decisions/experience-resolutions/values-with-growths-or-declines has results
[for non-Divine beings] that mold sanctifying-grace/life-essences into an accumulation of facet growths of said grace which give substance/life to the soul of that being so as to be henceforth eternal, since the substance [sanctifying grace] of ones faceted formation/self-definition/self-identification is itself eternal.
     A being/soul with no sanctifying-graced, facet accumulation is empty or 'dead' as an eternally existing 'aware, and essentially hollow, shell'.

3.  Decisions by finite beings are interposed among events and are non-predictable in any absolute sense,
but, when God makes decisions, they are infinite in capacity & understanding & all other regards, which includes that the decision provides for all variables for all future possibilities regarding anything & anyone [providing for any decisions, but not neccessarily knowing ahead of sequence what that personal decision will be] effected by that decision, which is just another way of saying that when God makes commitment to some course, then it is not done piecemeal, but all present decisions & future decisions for all variables are made at one occasion [but you still have personalized relationship with God on present time basis, but response from God was already provided for at beginning of time, and the substance of that decision was/is based upon your orientation towards that which God had already put in place, and with implementation by God or angels (for any necessary variables) or etc. on some particulars according to the exact nature of that present choice which you have enacted and according to your overall present set of values; and with this already arranged Godly response being an infinite exemplar of justice & love -- as well as being, as already specified, infinitely/eternally unchangeably set]; and which Godly set-of-decisions = Will = Person.

     God the Father = Will/set-of-decisions regarding commitment to creation & order [which includes process of 'being' creation & order]

     God the Son = Will/set-of-decisions regarding commitment to empower individual beings with some categories of said beings having free self-definition/self-identification/self-choice-of-facetvalues-makeup/image-of-God [with Christ having done likewise in chosing own makeup]
[God Himself entering into this process as archtype/summation/salvation (due to case of fall) of humanity = Christ]

     God the Paraclete/Holy Ghost = Will/set-of-decisions regarding commitment to God 'substance donational fractionation & fluctuation & interlink' with interlink/intercommunication of all fractions & within Trinity Itself.

4.  Necessity of the distinction that results from each Godly commitment/set-of-decisions, and so
necessity by human logic of the three different Personalities/Wills in the single Godhead's interface with creation is due to each set-of-decisions/Will-commitment/Person of Godhead being -- FROM THE POINT OF VIEW & OPERATION [PERSPECTIVE] OF CREATURES -- a unique process that embraces & enfolds & relates to all beings & all creation;
we interface with God differently under God's decision-process of creation/existence than we do with God as Paracletial grace-source, and again differently with God as Christ our Savior and Initiator of soul self-definition/image-of-Godness; and, especially, when such unique process of unique decision implementation & collection & memory is performed by some Being with unique location/relation-identification, we have what we as created beings define & identify as personality/person;
thus, there are three different commitment/decision-set processes, and with one decision-set process by Being Who often is multiply fractionated in substance, and one decision-set process by Being with known sometime physical location & form [form likely varies according to particular beings -- from other planets or galaxies or universes or dimensions -- with which Christ is relating] to His substance, and one decision-set process by Being Who is the continual & original remaindered source and integrational reference of that substance, and so we arrive at what is understood [according to human understanding] as identifying three different Persons;
accordingly, once the Godhead entered into [probably from the 'start'? of eternity, since it likely that these are inherent characteristics of the Godhead which It always displayed] Its three separate commitment processes [to us it is ongoing process, while to God it likely is completed decision-set], it was inevitable that we humans -- under Christ's encouragement -- would look upon [and could truly, and somewhat fully, understand -- but only by our logic] the Godhead and Its 3-way processed inter-face with us as being via three different  Persons [as we define beings, they are Persons];
that is, it is due to our sum of experiences and way of thinking that separate Personhood is rigthly [by our logic] identified/attached to the Godhead -- to God Himself there likely is no such necessity ["I and the Father are One"].

5.   Non-identicalness [according to human experience, and so human logic] pertains between being one with someone else, and being indistinguishable from them; all essentials of will/decision/action & nature & power & other essentials can be the same, while may still be non-identical/distinguishable by some aspect or another [such as awareness of commitment/decision-set with which are most closely identified by others/creatures].

6.   Oneness derives from essentials -- especially one's will & one's nature & one's awareness [if receive all information & experience of which all other humans are also aware, one would tend to form one best decision/appreciation together with everyone else, and so have a real sense of 'oneness' with all heavenly humanity -- the communion of saints].

7.   Harmony of all things is seen in reflection in natural universe of the same principles existing in God, and which principles 'required' the Trinity.
The three commitments/decision-sets/processes/Wills/Persons of God which are all needed together to form a 'relevant' universe [i.e. one with 'us' involved in it], coincides with the need for three reference elements to form structure [triangle] in the universe, and with these three always [eternally/infinitely] required or else never have structure;
and as the 'structure' in the universe on occasion  becomes operational system [tetrahedron/reality/physical-life] with addition of fourth reference element, so too the Godhead Trinity of 3 inherent infinities becomes a 'system of growth of soul' with addition of fourth potential infinite [collectively & over time/events] of created beings with the human/other grace-font-head/connection-to-ChristGod found in the Virgin Mary, Mother of ChristGod, to link each human soul to Christ's Divine Soul [without His linkage we have no access to Paraclete's essence/sanctifying-grace] and so to eternal life and standing in heavenly universe as partial children of the Divinity.