In Leviathan, the condition that men are in during the time they live "without a common power to keep them all in awe" is called...

So are you just asking what this time is called?  This
time is called the state of nature.  In the state of nature, there is what is often
called the war of each against all, though Hobbes' own words said that this war "is of
every man, against every man."

According to Hobbes, this is
why there is a need for a strong government.  If there is no strong government to keep
everyone in awe, the people will simply fight against one another all the time.  They
will constantly be trying to take each other's lands or possessions.  In this case, as
Hobbes' famous line goes, people's lives will be "solitary, poore, nasty, brutish, and
short."

So, Hobbes is saying that people will have terrible
lives if they do not have a strong government to keep them all in line.  They will be in
a perpetual state of war of every man against every man.