It's a body Wormtail created for him, through instructions Voldemort imparted.
"However, he was the able-bodied servant I needed, and, poor wizard though he is, Wormtail was able to follow the instructions I gave him, which would return me to a rudimentary, weak body of my own, a body I would be able to inhabit while awaiting the essential ingredients for true rebirth..."
As for the piece of Voldemort we see within Harry's "King's Cross" station, that was a dream for the most part. Dumbledore explains to Harry he was dreaming the whole situation and that he wasn't really there. I'd look at it as the way Harry's mind coped with the information he had been given plus being "killed". All the answers Harry was given in that sequence was information he already knew but never worked out before. Dumbledore was there because he had largely been his mentor and Harry was left for a long time without a real direction because Dumbledore died before getting the chance to explain what was ahead. The appearance of the piece of Volde's soul looks like the rudimentary Wormtail body probably for multiple reasons. Dumbledore tells him he can't help, even though he tries to help everyone, the feeble form is defenseless and needs help. The bigger reason is this is how Harry saw Voldemort before he got his body back, when their minds were accidentally linked, he would subconsciously believe a dying Voldemort looks this way.
Before encountering Quirrell, Voldemort had been inhabiting small creatures such as snakes. He found inhabiting creatures killed them pretty quickly and they had no magical abilities, so he was no better off than just being a bodyless soul.