I originally wrote this with the intention that it would be my first post in my Gospel Principles series, but it is not quite what I wanted to say, so I am going to rewrite. Still, consider this draft 1, and come back around January for Draft 2.
Before thoroughly discussing who our Heavenly Father is, it is essential to understand a few basic concepts that are fundamental to LDS theology. The first of these is the eternal nature of matter, and thus, by implication, the eternal nature of the reality we exist in. Thus it can be said our faith â€œholds strictly to the conception of a material universe.â€  Also, eternal laws bind this universe, and these laws bind anything within this universe.  
Also within reality are the eternal spirits of man. While it has been the subject of debate what state these spirits have always been in , we are bound by scriptural mandate to refer to our spirits as eternal, meaning they â€œhave no beginning; they existed before, they shall have no end, they shall exist afterâ€.  To take a leap, Iâ€™d assert this also implies another important feature of LDS belief, being that mankindâ€™s will has always been independent and free.
So basically, we have reality set up as this always-existent space, infinitely large, and infinitely full of material stuff, which is ruled by consistent laws of cause and effect and is full of eternal spirit beings.
This gets us to a beginning of sorts  with our Heavenly Father. Regarding LDS Cosmology, we may say that we have a correct idea of his character in that we trusted our Father in Heavenâ€™s knowledge and understanding of the eternal laws by which we are bound so much that we chose to be adopted by Him, to follow His plan, and to be His family.  God, seeing we lacked some essential characteristics He had  which somehow rendered us less than and other from Him, instituted a method of our being able to over come those deficiencies and improve, as He is. 
Further, Heavenly Fatherâ€™s interest is in our improvement, indeed our scriptures assert him stating this very thing to be so important to him that he has committed it to being the of his â€œwork and â€¦gloryâ€. This is because, as the Bible states, â€œGod is love.â€œ  [*]
So we have faith that God is superior to us in some fundamental way, but that He has the capacity and desire to help us to improve to become like Him. We also have faith that His method is correct and beyond reproach, in that Godâ€™s plan is in alignment with the universal, unbreakable laws of reality and thus cannot fail. God has a perfect understanding of these laws and their unchangeableness, as well as a perfect knowledge of us, His children. Lastly we know that our Father in Heaven chose to be called our Father, signifying not only his role as mentor, but the deep abiding and intimate interest He has in each of us, His children, and His commitment to help each of us individually as well as collectively. We believe this help is available to us now through various means, and that we can call out to our Father and receive His help via prayer. Not to over do it, but perhaps most radically, we believe God truly wants to help us by any means He can.  These things are fundamental and arguably universal in the LDS faith.
Other characteristics of our Father in Heaven have been discussed, but are subject to either further development later, or are speculative and debatable within the scope of LDS theology  While some may say with certainty they know more details than this about their Heavenly Father, I can not. I can say I believe in him, and I believe in His plan and in His love. It warms and strengthens me when I am weary and despondent. It gives me hope and purpose when I am lost in the throws of despair and apathy. I profess my faith and love for Him and declare his divinity. It is only fair; as my Fatherâ€™s love and faith in me are anchors I have available to me to shore me up every moment of the day. They are infinite pools of joy and peace from which I insatiably draw light and life. I do not know how I ever got by without them. 
 By Universe, I mean the whole of reality, and am not limiting to any subset thereof, I am not denying a multiverse per se, just simply being precise in that I aim to not only refering to the expanse of space and time generated by the big bang, but am thinking of space and time in terms of infinities.
 This would includeGod
 Abr 3:18
 Yes I know, no beginning, and all that, but Joseph Smith taught a beginning even for the God Head;
â€œAn Everlasting covenant was made between three personages before the organization of this earth and relates to their dispensation of things to men on the earth. These personages according to Abraham’s record are called God the first, the Creator; God the second, the Redeemer; and God the third, the Witness or Testatorâ€
[Extracts from Wm Clayton's Private Book, 10-11, Nuttall collection, BYU Library].
 How this adoption works is another subject that is often debated.
 One such common characteristic we gained in this life is a physical body. Latter-day Scripture boldly differentiates itself from its Christian contemporaries in teaching the essential nature of Godâ€™s body to his perfection and glory, and by extension as a critical component in manâ€™s search for happiness.
 For more detail on this, see where Iâ€™ve covered this before.
 Please forgive my not sighting what I think of as basic scriptures. If you are unfamiliar with them, just use the LDS scriptures search on lds.org
 For an idea of how radical this is, read Givensâ€™ beautiful essay on dialogic revelation here
 Examples include Godâ€™s foreknowledge, Godâ€™s sexuality, whether God is of the same â€œkindâ€ as man, Whether God is a single entity, a representative of a body of such beings, or is in fact a multiplicity of beings (Elohim does mean â€œGodsâ€ after all) Whether God has a female counterpart, whether Heavenly Father is in actuality the combination of a female and male divine being, and whether God went through a life as man does now.
 Obligatory link to conversion story here.
[*]- I removed the word entirety from this line, due to it being a possible overstatement of my position. Thanks A. Davis