What properties does Christianity attribute to God What are the claimed properties of God that define Christianity, and what is the basis or reason for each attribution?
This is a pretty broad question, and you have to understand that there are varying views within Christianity.  There's no one, single understanding, but here is a fairly comprehensive list.  Assume in this list that it applies to most groups within Christianity unless otherwise noted.  I'm going to ignore such groups a s "Christian Atheists" (Who would answer with one attribute: "Imaginary".) or "Christian witches" and focus on more mainstream commonly accepted Christian circles.

Also, in the interest of brevity, I'm going to NOT directly include "what is the basis" for each.  Instead, I'm linking on each to an article that explores each attribute in further detail.  The basis for each attribute is, ultimately, Scripture.  Most of the articles linked to offer several verses supporting each attribute.  




Omniscient:  God is all knowing, without limitation - even as far as past/present go.  He knows all that has ever happened and all that will.  He knows each star that he created, by name.
Omnipotent:  God is infinite in His power.  He can do anything.  


Some would say that this means Intrinsically possible.  Misunderstandings of the definition of "Omnipotent" lead to foolish questions like "Can God create a rock so big He can't lift it?"
Others believe that this limitation doesn't apply, and God id limited only by His own perfect nature ans will.

Omnibenevolent - God is purely good.  There is no evil in Him.  Note that this doesn't mean purely kind.  He is not limited to our personal understanding of what "good" is.  Put simply, God knows more than we, in our limited capacity do.  What we might consider "cruel" or "not fair" may be good.  Just like a child might not understand why his parents discipline him for dangerous behavior, we may not understand why God does what he does.





  Side note #1: Taking a break in the list here, many atheists use the above three attributes to come up with arguments that God couldn't
  possibly be all three, or evil wouldn't exist.  That's addressed
  here.





Sovereign:  God is completely sovereign.  he answers to nobody, and everybody answers to Him.


There is some debate on what this means for free will.  Some say that free will is an illusion because if we had any free will, Cod's sovereignty would be nullified  In other words, He is in complete and total control.  Others say that He grants us free will, but he could assert His sovereignty at any time.  There are other views as well.  
Some also use god's sovereignty to show that God is evil because He allows suffering.  Again, see side note #1.

Infinite:  God is without limits.  He is limited only by His will and what is intrinsically possible.
Patient (aka longsuffering):  God is slow to anger, quick to forgive.  Which is good, because when I look at my own behavior over my lifetime, I can't help but think that, were I in God's place, I'd have been a crispy critter long ago.
Holy: God is separate from sin and is completely, utterly, incorruptible.
Impeccable:  Strongly related to holiness, God is completely unable to sin.
Immutable: God Cannot change.  


Side note, skeptics sometimes point to verses that showed that God repented and decided not to smite this group or that.  That's addressed here.

Omnipresent: God is everywhere, always, all at once.
Transcendent: God exists outside space and time or anything else he has made.  he is not limited by either.
Immanent: Closely related to "Omnipresence":  God is always near.
Jealous:  God is jealous of glory due Him being given to another god, and righteously so.  This is to be distinguished from petty jealousy (envy).
Righteous: God is completely Right, or Just.  It is His righteousness that demands that sin not go unpunished.  (Would you consider a judge to be righteous if he let a murderer go free out of love?  No, you'd call that a corrupt judge.  in the same way, God cannot be righteous if he allows sin to go unpunished.)
Wrathful:  God's wrath, when unleashed, is unparalleled. His wrath is directed toward sin, evil, unrighteousness.
Love:  While God's wrath is terrifying, his love is amazing.  It is his love that is expressed in the fact that while we were yet sinners, Christ died for us.  The cross is the ultimate expression of his love for us.
Incorporeal: Scripture tells us that god is spirit, and has no physical body.


Note that this is something that some groups, such as the LDS church disagree with.

Triune: God exists in three persons: The Father, the Son, and the Holy Spirit.


Pretty much undisputed in mainstream Christianity, understood differently in LDS theology, and refuted utterly by other groups such as Jehovah's Witnesses.



And finally....


Incomprehensible: God is so far above us that we can never know or understand Him fully.  All we can know or understand is what He has revealed, and even that is filtered through our own interpretations.