I would like to ask, when could the common ancestor of insects and people (vertebrates) live? How could it look like?

This division represents one the major splits in kingdom animalia. Insects belong to phylum arthropoda, which contains many other highly diverse groups like spiders, crustaceans and centipedes. Arthropoda is arguably one of the most successful animal groups on the planet. Vertebrates belong to phylum chordata, another highly diverse and successful group.

The diversion is thought to have happened on the order of hundreds (~500) of millions of years ago.


Hello just to add to the answer that John has given you, the split
between us and insects is the split between the animal groups called
protostomes and dueterostomes.  These names refer to when the mouth
develops during the formation of the gut; mouth first in protostomes,
while the deuterostomes' mouth secondarily, as the anus develops first. 
Bilaterally symmetrical animals are first found in the fossil record
during the Cambrian (~542-490 Million Years Ago).  We have both
protostomes and deuterostomes (arthropods and chordates- if not actual
vertebrates) are present in Chinese fossil beds, like the Chengjiang
Formation.  This bed is about 530 million years old, so the last common
ancestor of arthropods and chordates has to be older than this. 
oldest definitive animal fossils that we have are the embryos from the
Doushantuo Formation also from China.  The Doushantuo is 581 million
years old.  The embryos are 'sponge-like'  so they represent the most
basal animal body form.  The oldest fossil does not (necessarily)
represent the first individuals of a group, only the 'first' to be
preserved, but it is a reasonable estimate for the purposes of your
question to give a range of times when the last common ancestor of
bilaterians might have appeared.  We have about 50 million years
(581-530 MYA)  between the appearance of animals (sponge-like) in, and
the appearance of arthropods and vertebrates in, the fossil record, for
the last common ancestor to have existed.  The second part of your
question is very important to developmental and evolutionary biologists,
because understanding the body plan of an ancestor gives us more
ability to understand how animals develop and evolve.  The way that we
start to get an idea of what an ancestor looks like is based in part on
what its descendants look like.  More importantly, the shared
characteristics of different groups 'hint' at the characteristics of the
ancestor.  For example, vertebrates, arthropods and annelids are
segmented (and although this is contentious) the Last Common Ancestor
(LCA) was probably also segmented, at the very least it likely showed a
banding pattern of gene expression diving up the body.  It had a through
gut a mouth at one end  and an anus at the other.  It had sense organs,
eye spots patterned by a gene called Pax-6.  To give an idea of what it
might have been, I think that the best description is 'kind of
worm-like'.  I hope that this helps to answer your question.