I'll try to explain briefly.
"Heaven's feel" is just another term for Holy Grail War. It's worth noting that the japanese term has a different meaning and "heaven's feel" isn't a translation but something the author himself made up.
the Great Holy Grail is the magic circle inside the cave which Shirou destroys in the HF endings. It's purpose is to slowly collect mana from the region's leylines and when it's ready, it will automatically pick masters and give them the ability to summon a servant. And so a Holy Grail War begins.
The power to summon servants is a miracle in itself but the instigators of the ritual were after an omnipotent power, and that's what the actual Holy Grail is for. The souls of killed servants are collected within a magus specialised for that purpose, usually a homonculus created by the Einsberns (Ilya) but other options are also explored in the story (Sakura, made compatible by Zouken and Shinji with Ilya's heart). Anyway, when the magus has enough souls, he/she can be used to summon the Holy Grail in one of the designated ritual sites, such as the temple or the church grounds.
The reason the Holy Grail has become so nasty is that a few wars back, Angra Manyu was summoned and he somehow integrated himself with the grail. He wasn't a very heroic spirit, so now all the grail is good for is death, destruction, burning, tentacle rape and making obscene red text appear in your head.