Many people have difficulty understanding the three-in-one God.
A few analogies have been developed over time in an attempt to clarify. The most helpful to my knowledge is the mathematical one: 1 x 1 x 1 = 1.
Others are the three sides of a triangle, the three states of matter (ice, steam and water are all water, but this reeks of modalism).
The triangle almost makes it, but I prefer the three legged stool in Flatland (a two-dimensional world). In Flatland the three legged stool is just the three disks where the legs touch Flatland. You move one disk, and the other two follow, because they are part of the one object despite it not being visible in Flatland, and they still maint their relationship, existence and identity. Touch any one of them, and you touch the stool. Yet, they are three individual disks. One stool in three disks.
But I think a closer analogy is the family.
I'm husband Jones, my wife is wife Jones and our daughter (who proceeds from the husband (father) and wife (mother) is daughter Jones. We are all Joneses. Any one of us is fully a Jones.
The intimacy we have in a family is but a faint hint of the unity of God. God is indeed one. He also exists in three real individual persons. Their representation in scripture is clear on that.
The unity of God is that all three persons know each other with comprehensive intimacy and speak, act and know in that indissoluble unity.
A husband and wife mimic this a little: they become of one flesh; but God is of one god-ness. Daughter Jones is of the same flesh as her parents. Translate this to the spiritual realm and you might have it: God is one in divinity while three in persons all unified in intent, being and existence.