How to name the compound with the formula Br3C-CHCl2?
The first name is correct (1,1,1-tribromo...).  Counting the substituents from either end, you would get 1 for the first substituent going from either direction.  As a tie-breaker, look at the next substituent, which would give you 1,1 going from the bromine end, and 1,1 going from the chlorine end.  Since the tie is still not broken, you go to the third substituent.  Going from the bromine end you would have 1,1,1 but 1,1,2 going from the chlorine side.  So, you number the carbon with the bromines number 1, giving you 1,1,1-tribromo-2,2-dichloroethane.

As a side note, if the carbon on the right had three chlorines instead of two, this method wouldn't work (you'd get 1,1,1,2,2,2 numbering from either direction).  In this circumstance you would use alphabetical order to break the tie, with bromo- having higher priority than chloro.  That would give you 1,1,1-tribromo-2,2,2-trichloroethane.  I know you didn't ask this, but it could prove useful to know!