You cannot arbitrarily add a variable into a different function unless you are passing it in, or it is global.
>>> def foo():
...     oof = 0
... 
>>> def fo():
...     oof+=1
... 
>>> foo()
>>> fo()
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "<stdin>", line 2, in fo
UnboundLocalError: local variable 'oof' referenced before assignment
>>> 
This causes an error because oof is not defined in fo(). One way to change that is to add global variables:
>>> def foo():
...     global oof
...     oof = 0
... 
>>> def fo():
...     global oof
...     oof+=1
... 
>>> foo()
>>> oof
0
>>> fo()
>>> oof
1
>>> 
You can also fix the error by passing in the variable:
>>> def foo():
...     oof = 0
...     return oof
... 
>>> def fo(oof):
...     oof+=1
...     return oof
... 
>>> oof = foo()
>>> oof
0
>>> oof = fo(oof)
>>> oof
1
>>> 
Global variables in your code:
def startup():
        print "Welcome to the Text Based Hockey League!"
        print "You will be tasked with being a team's General Manager"
        global yourTeam
        yourTeam = raw_input()
class tbhl:
    def __init__(self):
        global yourTeam
        self.teamList["Mustangs", "Wolves", "Indians", "Tigers", "Bears", "Eagles", yourTeam]
class game:
    def __init__(self, homeScore, awayScore):
        #games left in a team class
        pass
startup()
mylist = tbhl()
print tbhl.teamList[6]
Passing in variables in your code:
def startup():
        print "Welcome to the Text Based Hockey League!"
        print "You will be tasked with being a team's General Manager"
        yourTeam = raw_input()
        return yourTeam
class tbhl:
    def __init__(self, yourTeam):
        self.teamList["Mustangs", "Wolves", "Indians", "Tigers", "Bears", "Eagles", yourTeam]
class game:
    def __init__(self, homeScore, awayScore):
        #games left in a team class
        pass
yourTeam = startup() #<-The line that the error points to
mylist = tbhl(yourTeam)
print tbhl.teamList[6]