"We believe in God the Eternal Father, and in his Son Jesus Christ and in the Holy Ghost." 
 
We believe that they are 3 distinct and seperate beings. 
 
Heavenly Father is our father, and we are his Children. (Duh, I hope that's not a revelation for you, or if it is, Welcome aboard! : )) The easiest way for me to think of it is that I am in resemblance of my Father. I think it would be a little weird for me to think that my father didn't have a body like I did, and he didn't know, and actually experience what it was like to have one. 
 
So Ya, God has a body. But more than that. If you go and read in the other Questions, one asks if we actually believe that we can become Gods. Again, the answer is Yes. We believe that progression is an ongoing thing throughout eternity. But more than that too, We believe that God is a Ressurected Being. That He once had the opportunity to experience what we are, to have a physical body and he lived as we do now. But he overcame everything that was required of him, and because of that was able to progress to where he is now.  
 
A very simplified answer though, is that Joseph Smith (Who we believe to be the first in a continuing line of many "Latter Day" Prophets down to the current time) Saw him, and saw that he did.